I know that their are other versions of this story but these two screenshots are from Vayu Purana pages 101 and 104. No version of this story mentioned Kartavirya needing to build a bridge to get to lanka. Also Vayu Purana doesn't even mention Kartavirya needing an army to attack lanka (other versions say he needed an army during the battle). Why did Rama need an army? Why did Rama need to build a bridge?
Yes kartavirya-arjuna or Sahastra Bahu Arjuna was very powerful Kshtriya Warrior. He could have inveded Lanka alone & defeated Ravana alone , without even need of an army or bridge on sea to reach the lanka ,due to his powers and siddhis.
Now let's try to answer your questions from scriptures. First of all let's see what kartavirya-arjuna was capable of and what powers he was possessing. His powers and the boons he got from Shree Dattatreya are mentioned in Shreemad Bhagvat Purana -Skanda 9 -Chapter 15- Verses 17-18-19.
श्री शुक उवाच
हैहयानामधिपतिरर्जुनः क्षत्रियर्षभः ।
दत्तं नारायणांशांशमाराध्य परिकर्मभिः ॥17॥
बाहून् दशशतं लेभे दुर्धर्षत्वमरातिषु ।
अव्याहतेन्द्रियौजः श्री तेजोवीर्ययशोबलम् ॥18॥
योगेश्वरत्वमैश्वर्यं गुणा यत्राणिमादयः ।
चचाराव्याहतगतिर्लोकेषु पवनो यथा ॥20॥
haihayānām adhipatir arjunaḥ kṣatriyarṣabhaḥ dattaḿ nārāyaṇāḿśāḿśam ārādhya parikarmabhiḥ bāhūn daśa-śataḿ lebhe durdharṣatvam arātiṣu avyāhatendriyaujaḥ śrī-tejo-vīrya-yaśo-balam yogeśvaratvam aiśvaryaḿ guṇā yatrāṇimādayaḥ cacārāvyāhata-gatir lokeṣu pavano yathā
Sukadeva Gosvami said: The best of the kshatriyas, Kartaviryarjuna, the King of the Haihayas, received one thousand arms by worshiping Dattatreya, the plenary expansion of the Supreme Personality of Godhead, Narayana. He also became undefeatable by enemies and received unobstructed sensory power, beauty, influence, strength, fame and the mystic power by which to achieve all the perfections of yoga, such as anima and laghima. Thus having become fully opulent, he roamed all over the universe without opposition, just like the wind. SB 9.15.17-19
So he was undefeatable by enemies , was possesing enormous strength , His bodily strength i.e. was inexhaustible.Besides he was also possessing Yog-Siddhis like anima and laghima. With which he could have easily defeated Ravana of Lanka.
Also note that as said in the Vayu Purana quote in your question ,kartavirya-arjuna did not invaded lanka and made Ravana captive from bringing him from there but the story took place at banks of River Narmada.
Now Let's see what Shree Rama Was capable of..
According to Bhagvat Purana and Valmiki Ramayana. Shree Rama was avatara of Lord Vishnu. In Rama-centric traditions of Hinduism, he is considered the Supreme Being. i.e. ParaBrahman.
As a person, Rama personifies the characteristics of an ideal person (purushottama). He was Maryada Purusottama and His kingdom was called Rama-Rajya.
The very precise answer of your query is given in this verse of Bhagvatam.
It's said in the verse that Lord Rama did not needed any help of monkeys or didn't needed to build bridge on sea , It's all the leela of Bhagavanta. As Lord himself took this avatara upon request of god's. The power of Lord is supreme against any normal human being like kartavirya-arjuna ,who were killed by Avatara of Lord himself . When arjuna forcibly took kamadhenu from Rishi jamadagni.
नेदं यशो रघुपतेः सुरयाच्ञयात्त
किं तस्य शत्रुहनने कपयः सहायाः ॥20॥
Lord Ramacandra’s reputation for having killed Ravana with showers of arrows at the request of the demigods and for having built a bridge over the ocean does not constitute the factual glory of the Supreme Personality of Godhead Lord Ramacandra, whose spiritual body is always engaged in various pastimes. Lord Ramacandra has no equal or superior, and therefore He had no need to take help from the monkeys to gain victory over Ravana. SB 9.11.20
Simply Rama did not needed to bulid bridge nor he was requiring any help of Monkeys .