Someone who is hospitalized over Shabbos in a hospital where the kashrus supervision isn’t up to standard, is there a way he can separate terumos and maasros on Shabbos

Yes. One can separate terumos and maasros on his tevel produce by making a t’nai.  A t’nai means saying the entire nusach of separating terumos and maasros before Shabbos, but in future tense (for the full nusach as such, see Mishpetei Aretz 19:4). Then, on Shabbos, one would separate teruma and maaser normally.

However, one cannot separate terumos and maasros with a t’nai if he doesn’t own the produce. One can argue that in a hospital setting the fruits are in fact considered “his” since by law the hospital has to provide the patient with food. Nevertheless halachically he’s not considered the owner of the produce. Therefore, HaRav Elyashiv זצ”ל ruled that the patient should go to the mashgiach in charge at the hospital, since he is the hospital’s representative, and should ask his permission to separate teruma and maaser (i.e., as his shaliach). He can then make the t’nai before Shabbos and separate teruma and maaser on Shabbos.

If he’s in a hospital that has a good hechsher on the food but wants to separate terumos and maasros anyhow as an extra hidur, he is allowed to separate trumos and maasros if he made a t’nai before Shabbos.