If a person does daily "hitbodedut" meditation, what is he allowed to pray for on Shabbos?

On Shabbat, it's better to concentrate on praising God and His world, and thanking Him for everything in joy, and not focus on what we're lacking and need, which may make you sad. If it's important for you to ask for something, you can ask for those things we ask for in the prayers of Shabbat (e.g. at the end of Shmoneh Esreh, in Ahava Rabba before Shma Yisrael, in the "Mi sheBerachs of the olim to the Torah, for the State of Israel, the Y'kum Purkans, etc.) which the rabbis excepted from the aforementioned rule.