In shaarey teshuva one category of sin is punishible by mitah b’dey shamayim, bar minan, however in later chapters where teshuvah is broken up in categories, there are categories for teshuva for aseh, lo taaseh, karet, mitah m’bet din shel mateh, but i did not see anything for teshuva for mitah b’dey shamayim; what is the teshuva and reprecussions for this? (see also perhaps rambam hilchos teshuva)
The Shaarei Tesuvah mentions misah bidey shamayim as a category of sin (3:107 and on), and mentions various sins that are punishable by misah bidey shamayim. However, as the question notes, he does not state any special method of teshuvah for this category. Rather, the teshuvah required will depend on the category into which the sin can be slated.
For example, one of the sins mentioned is a Torah scholar who acts immodestly, which causes a chilul Hashem (3:113); this will fit into the later category of chilul Hashem. Another example is motzi shem ra, which involves a negative commandment, and fits into that category.