Both the Arjunas (the King of Haihaya, and the other is the grandfather of the child) are famous for their bowmanship, and the child Pariksit is foretold to be equal to both of them, particularly in fighting
SB Canto 1
Amongst great bowmen, this child will be as good as the Arjunas. He will be as irresistible as fire and as unsurpassable as the ocean.
In history there are two Arjunas. One is Kārttavīrya Arjuna, the King of Haihaya, and the other is the grandfather of the child. Both the Arjunas are famous for their bowmanship, and the child Parīkṣit is foretold to be equal to both of them, particularly in fighting. A short description of the Pāṇḍava Arjuna is given below:
Pāṇḍava Arjuna: The great hero of the Bhagavad-gītā. He is the kṣetra-ja son of Mahārāja Pāṇḍu. Queen Kuntīdevī could call for any one of the demigods, and thus she called Indra, and Arjuna was born by him. Arjuna is therefore a plenary part of the heavenly King Indra. He was born in the month of Phālguna (February-March), and therefore he is also called Phālguni. When he appeared as the son of Kuntī, his future greatness was proclaimed by air messages, and all the important personalities from different parts of the universe, such as the demigods, the Gandharvas, the Ādityas (from the sun globe), the Rudras, the Vasus, the Nāgas, the different ṛṣis (sages) of importance, and the Apsarās (the society girls of heaven), all attended the ceremony. The Apsarās pleased everyone by their heavenly dances and songs. Vasudeva, the father of Lord Kṛṣṇa and the maternal uncle of Arjuna, sent his priest representative Kaśyapa to purify Arjuna by all the prescribed saṁskāras, or reformatory processes. His saṁskāra of being given a name was performed in the presence of the ṛṣis, residents of Śataśṛṅga. He married four wives—Draupadī, Subhadrā, Citrāṅgadā and Ulūpī—from whom he got four sons of the names Śrutakīrti, Abhimanyu, Babhruvāhana and Irāvān respectively.
During his student life he was entrusted to study under the great professor Droṇācārya, along with other Pāṇḍavas and the Kurus. But he excelled everyone by his studious intensity, and Droṇācārya was especially attracted by his disciplinary affection. Droṇācārya accepted him as a first-grade scholar and loved heartily to bestow upon him all the blessings of military science. He was so ardent a student that he used to practice bowmanship even at night, and for all these reasons Professor Droṇācārya was determined to make him the topmost bowman of the world. He passed very brilliantly the examination in piercing the target, and Droṇācārya was very pleased. Royal families at Maṇipur and Tripura are descendants of Arjuna's son Babhruvāhana. Arjuna saved Droṇācārya from the attack of a crocodile, and the Ācārya, being pleased with him, rewarded him with a weapon of the name brahmaśira. Mahārāja Drupada was inimical toward Droṇācārya, and thus when he attacked the Ācārya, Arjuna got him arrested and brought him before Droṇācārya. He besieged a city of the name Ahichhatra, belonging to Mahārāja Drupada, and after taking it over he gave it to Droṇācārya. The confidential treatment of the weapon brahmaśira was explained to Arjuna, and Droṇācārya was promised by Arjuna that he would use the weapon if necessary when he (Droṇācārya) personally became an enemy of Arjuna. By this, the Ācārya forecasted the future battle of Kurukṣetra, in which Droṇācārya was on the opposite side. Mahārāja Drupada, although defeated by Arjuna on behalf of his professor Droṇācārya, decided to hand over his daughter Draupadī to his young combatant, but he was disappointed when he heard the false news of Arjuna's death in the fire of a shellac house intrigued by Duryodhana. He therefore arranged for Draupadī's personal selection of a groom who could pierce the eye of a fish hanging on the ceiling. This trick was especially made because only Arjuna could do it, and he was successful in his desire to hand over his equally worthy daughter to Arjuna. Arjuna's brothers were at that time living incognito under agreement with Duryodhana, and Arjuna and his brothers attended the meeting of Draupadī's selection in the dress of brāhmaṇas. When all the kṣatriya kings assembled saw that a poor brāhmaṇa had been garlanded by Draupadī for her lord, Śrī Kṛṣṇa disclosed his identity to Balarāma.
He met Ulūpī at Haridvāra (Hardwar) and was attracted by that girl, who belonged to Nāgaloka, and thus Iravān was born. Similarly, he met Citrāṅgadā, a daughter of the King of Maṇipura, and thus Babhruvāhana was born. Lord Śrī Kṛṣṇa made a plan to help Arjuna to kidnap Subhadrā, sister of Śrī Kṛṣṇa, because Baladeva was inclined to hand her over to Duryodhana. Yudhiṣṭhira also agreed with Śrī Kṛṣṇa, and thus Subhadrā was taken by force by Arjuna and then married to him. Subhadrā's son is Abhimanyu, the father of Parīkṣit Mahārāja, the posthumous child. Arjuna satisfied the fire-god by setting fire to the Khāṇḍava Forest, and thus the fire-god gave him one weapon. Indra was angry when the fire was set in the Khāṇḍava Forest, and thus Indra, assisted by all other demigods, began fighting with Arjuna for his great challenge. They were defeated by Arjuna, and Indradeva returned to his heavenly kingdom. Arjuna also promised all protection to one Mayāsura, and the latter presented him one valuable conchshell celebrated as the Devadatta. Similarly, he received many other valuable weapons from Indradeva when he was satisfied to see his chivalry.
When Mahārāja Yudhiṣṭhira was disappointed in defeating the King of Magadha, Jarāsandha, it was Arjuna only who gave King Yudhiṣṭhira all kinds of assurances, and thus Arjuna, Bhīma and Lord Kṛṣṇa started for Magadha to kill Jarāsandha. When he went out to bring all other kings of the world under the subjection of the Pāṇḍavas, as was usual after the coronation of every emperor, he conquered the country named Kelinda and brought in subjugation King Bhagdutt. Then he traveled through countries like Antagiri, Ulukpur and Modapur and brought under subjugation all the rulers.
Sometimes he underwent severe types of penances, and later on he was rewarded by Indradeva. Lord Śiva also wanted to try the strength of Arjuna, and in the form of an aborigine, Lord Śiva met him. There was a great fight between the two, and at last Lord Śiva was satisfied with him and disclosed his identity. Arjuna prayed to the lord in all humbleness, and the lord, being pleased with him, presented him the paśupata weapon. He acquired many other important weapons from different demigods. He received daṇḍāstra from Yamarāja, paśāstra from Varuṇa, and antardhana-astra from Kuvera, the treasurer of the heavenly kingdom. Indra wanted him to come to the heavenly kingdom, the Indraloka planet beyond the moon planet. In that planet he was cordially received by the local residents, and he was awarded reception in the heavenly parliament of Indradeva. Then he met Indradeva, who not only presented him with his vajra weapon, but also taught him the military and musical science as used in the heavenly planet. In one sense, Indra is the real father of Arjuna, and therefore indirectly he wanted to entertain Arjuna with the famous society girl of heaven, Urvaśī, the celebrated beauty. The society girls of heaven are lusty, and Urvaśī was very eager to contact Arjuna, the strongest human being. She met him in his room and expressed her desires but Arjuna sustained his unimpeachable character by closing his eyes before Urvaśī, addressing her as mother of the Kuru dynasty and placing her in the category of his mothers Kuntī, Mādrī and Śacīdevī, wife of Indradeva. Disappointed, Urvaśī cursed Arjuna and left. In the heavenly planet he also met the great celebrated ascetic Lomasa and prayed to him for the protection of Mahārāja Yudhiṣṭhira.
When his inimical cousin Duryodhana was under the clutches of the Gandharvas, he wanted to save him and requested the Gandharvas to release Duryodhana, but the Gandharvas refused, and thus he fought with them and got Duryodhana released. When all the Pāṇḍavas lived incognito, he presented himself in the court of King Virāṭa as a eunuch and was employed as the musical teacher of Uttarā, his future daughter-in-law, and was known in the Virāṭa court as the Bṛhannala. As Bṛhannala, he fought on behalf of Uttara, the son of King Virāṭa, and thus defeated the Kurus in the fight incognito. His secret weapons were safely kept in the custody of a somi tree, and he ordered Uttara to get them back. His identity and his brothers' identity were later on disclosed to Uttara. Droṇācārya was informed of Arjuna's presence in the fight of the Kurus and the Virāṭas. Later, on the Battlefield of Kurukṣetra, Arjuna killed many great generals like Karṇa and others. After the Battle of Kurukṣetra, he punished Aśvatthāmā, who had killed all the five sons of Draupadī. Then all the brothers went to Bhīṣmadeva.
It is due to Arjuna only that the great philosophical discourses of the Bhagavad-gītā were again spoken by the Lord on the Battlefield of Kurukṣetra. His wonderful acts on the Battlefield of Kurukṣetra are vividly described in the Mahābhārata. Arjuna was defeated, however, by his son Babhruvāhana at Maṇipura and fell unconscious when Ulūpī saved him. After the disappearance of Lord Kṛṣṇa, the message was brought by Arjuna to Mahārāja Yudhiṣṭhira. Again, Arjuna visited Dvārakā, and all the widow wives of Lord Kṛṣṇa lamented before him. He took them all in the presence of Vasudeva and pacified all of them. Later on, when Vasudeva passed away, he performed his funeral ceremony in the absence of Kṛṣṇa. While Arjuna was taking all the wives of Kṛṣṇa to Indraprastha, he was attacked on the way, and he could not protect the ladies in his custody. At last, advised by Vyāsadeva, all the brothers began their mahā-prasthāna. On the way, at the request of his brother, he gave up all important weapons as useless, and he dropped them all in the water.