Practically it would seem, not only does the bochur not need to travel to another city to ensure he can daven with a minyan on Yom Tov Sheini, on the contrary, it’s better for him to stay in yeshivah. The Rema (529:1) writes: It’s forbidden to eat on erev Yom Tov after the time of mincha ketanah like on erev Shabbos. This is due to kavod Yom Tov, so that one should have appetite on Yom Tov night.
The Biur Halachah brings from the Magen Avraham that on first day Yom Tov as well, one should refrain from eating after nine hours into the day, since it’s erev Yom Tov Sheini and one needs to make sure to have an appetite on the night of Yom Tov Sheini. The Biur Halachah asks: Since today we have a fixed calendar, to be stringent about eating on first day Yom Tov because of the second day, especially on a Yom Tov such as Shavuos when there is still plenty time left in the day, and to make out as if the first day is a sofek [doubt] and treat it like weekday so that one can eat with appetite on second day Yom Tov, where is there basis for such a thing?
In short, the Biur Halachah is asking: First day Yom Tov is at least as important as second day, so why should one minimize simchas Yom Tov on the first day just because of the second – especially today when we have a fixed calendar, and the main day of Yom Tov is the first?
Based on this Biur Halachah, in our case as well, chas vesholam to be mezalzel [degrading] in Yom Tov Sheini, however, to reduce in simchas Yom Tov on the first day of Yom Tov by leaving yeshiva etc. just for second day Yom Tov, and to obligate the bochur already on erev Yom Tov, to go to another city because of Yom Tov Sheini, where do we find basis for such a thing? Therefore, it would seem that it’s better for him to stay in yeshivah and learn with his chavrusas etc. and not worry about travelling to another city in order to have a minyan and krias haTorah on Yom Tov Sheini.