Let's try to break the sentence into parts.
突然 can be called a の-adjective (more on that here). There are myriad connections that の can perform (not just the usual possessive one). If you're interested in all the possible meanings of の, I recommend checking the book A Dictionary of Basic Japanese Grammar.
転校 is a name, which is being qualified by 突然. All in all, the group 突然の転校 is a name, meaning "sudden change of school".
どうしようもなく comes from どうしようもない, which means "there's nothing that can be done about it". ない is an adjective, and although the く form normally changes the adjective into an adverb, it can also be used to connect sentences (more on this here), similar to the て form.
As a side note, the stem of a verb can also be used to connect sentences.
From here, it's easier to grasp the meaning of で: it's connecting the two sentences. For nouns and な adjectives, you used で for connections such as 賑やかで大きな町 (a lively and big town).
Summarising, the sentence means
There's nothing I can do about the sudden change of schools. (lit. there's a sudden change of schools and there is nothing to be done about it)
Edit:
The previous example connected two な adjectives, but it can be equally used to connect sentences/ideas (as is the case in the OP sentence). Here's a better example, taken from a college website:
やましたせんせいはにほんじんで、よんじゅっさいぐらいです。