In my opinion Arjun's first exile was 12 months.but the second exile that all.Pandaavs went to was 13 years.
Why was Arjun's first exile 12 months. Arjun was the primary protector of Pandavs when it came to fighting armies. Bhima was the protector in terms.of single enemies. That is why Arjun was commander and Bhima like a body guard.
Like I mentioned earlier, only After Rajasuya Yudhishtir became an.independent king in his own merit. Till then he was just a extended dominion of Hstinapur, but without the protection of Hastinapur. He was at the mercy of Panchal army. He had Jarasandh baying for his blood for siding with Krishna and he had treacherous Duryodhan onnthe other side.
On top of it Robbers were looting and plundering. Nagas had the entire khandav forest in their control denying any access to Pandava. Most importqntly Pandavas did not have a well trained battle ready army of warriors of their own
IP was in a very precarious position for Arjun to go gallivanting for 12 years.
If you forget the exile part and only look objectively Arjun does 4 things in this exile. He succeeds in 3, but fails in the fourth
1. Has a brief encounter with naga ulupi who happens to be of Kouravya clan. Kunti's biological mother was also from the same clan. So Arjun is technically reaxhing out to relatives for brokering peace with Nagas at Khandavvan
2. Marries Chitrangada from manipur. The name itself would tell you what Manipur must be famous for. Krishna built dwarka with the help of syamantak Mani. Precious gems and jewellery from manipur would help fill the coffers at IP nicely.
3. Then he tries to meet Parshuram but is unable to do so as he has gone somewhere else. This must be to gain some divine weapons from him. But this does not happen.
4. Next he meets Krishna and marries his sister and cements the formal alliance with Dwarka, which was only informal so far. From here as well he gets a huge dowry for Indraprasth.
He had no plans of roaming here and there. He soecifically went to places planned with clear intent
If he left on day of the year, he spent 15 days in sealing the deal with Ulupi's clan.Then left for Manipur. He marries Chitra and leaves to meet Parshuram as soon as she conceives. Parshuram is a no go so returns back, stays for the baby!s birth and in the 11th month reaches dwarka straight. Waits a couple of weeks and then kidnaps subhadra marries her and probably spends a month more and returns back.
He had gone for specific tasks. For eg. The kings who came.for draupadi swayamvar from distant corners of India, would they have taken years to reach Kampilya. No. They were coming for a specific event. They would have reached the place within weeks if not days.
But the exile after Dice game has to be 13 years.
Reason 1. Abhimanyu , Ghatothkach Iravan and Uttara would be toddlers.
Reason 2. The kings who supported Pandavas were the second generation guys who had just gained the thrones in their own kingdom. None of them were in a position to fight at the time. Shishupal had just died leaving behind a lot of enemies and Dhrishtaketu came to power. Jarasandh had died and his son Sahdev was fighting his own battles in magadha. Salwa had severly damaged dwarka. Kekayas were having their own problems. Uttara was toddler who could not be married immediately to seal matsya pandav alliance. Panchalas I believe may have been ok. But they were the only ones then.