If a person has Rebeinu Tam Tefilin (Beit Yosef) and finds himself in a situation without his Tefilin. Would he fulfill the chumra of Rebeinu Tam Tefilin by using someone else’s which are Ktav Ari.
Yes, according to most authorities, Ketav Ari is fine (bedieved) for fulfilling mitzvos related to setam, and this is certainly the case for the chumra of Rabbeinu Tam tefillin.
Although some authorities (in particular the Chazon Ish, who comments on the inverted tzaddik) cast doubt over whether one can fulfill mitzvos of setam with Ketav Ari, the majority of authorities rule that this is okay, at least on a bedieved level. It is certainly fine to rely on this for purposes of Rabbeinu Tam tefillin.
See Shaarei Binah (Shiloni, a very thorough work on the subject), kelal 5, who cites the relevant sources.