I refer you to the top comment in this /r/linguistics thread but the short answer is 'no, not really', there is some scope for vague guesswork, based on stuff like assuming parallels with other proximal languages and dialects (e.g Akkadian) but that's about it.

I just wanted to point out that, although the Gilgamesh epic does go back as far as the Sumerian period, the specific version of Tablet V that was recently discovered with previously-missing lines was actually written in Neo-Babylonian Akkadian, not Sumerian. You can read all about it here.

As for Sumerian phonology, like the thread in /r/linguistics suggests, we don't really have a very clear idea on how it would've been pronounced. Scholars can make guesses on the basis of linguistic derivation in Akkadian dialects, but that's an uncertain process at best. Even the number of vowel sounds is disputed; at least four, with the basic "a, i, e, u", but perhaps five or as many as eight.