Okay, I grew up with this song - my mom Ukrainian and father Russian and constantly heard it on Red Army Chorus albums. This is the best I can do phonetically with my knowledge of Russian.
Eyy, ookh-nyem! (eyy as in "play", but sung ei-yoohkhnyem) Eyy, ookh-nyem! Ye-shcho raz ee, ye-eshcho-da raz! (ye as in yes)
Ra-zo-vyom miy ber-yo-zoo, (mi, the "i" being short but with an almost silent y at the end as in "miue" Raz-o-vyom miy koo-drya-voo, Ai da da, ai da, ai da da, ai da, Ra-zo-vyom miy koo-drya-voo.
Chorus
Miy po byer-yezh-koo i-dyom, Pye-snyu sol-nish-koo poy-yom... (sol as is "soul") Ai da da, ai da, ai da da, ai da, Pye-snyu sol-nish-koo poy-yom... Chorus
Ekh, tiy, Vol-ga, ma-tryeka, Shee-ro-ka ee gloo-bo-ka... Ai da da, ai da, ai da da, ai da, Shee-ro-ka ee gloo-boka...
all r's are pronounced with a roll - rrrrrr!