It's not the best source to learn modern Russian in that sense that for obvious reasons one won't find any useful, relevant and colloquial phrases in Bible. This, however, actually holds true about the New Russian Translation.
As of how comprehensible is Synodal Bible - it's actually very understandable and native speaker won't find it too difficult to understand. But, once again, keep in mind - comprehensible and modern are two different thing. The lexicon is still archaic. Nobody says "нарёк" now but rather "назвал", instead of "убоялся" you'll encounter rather "испугался", nobody says "вчера был в клубе, и познал одну горячую девицу" etc.
So if somebody will thoroughly and systematically treat Synodal Bible as one and only source of learning Russian it won't be a tragedy however I'd rather recommend not to do it.
By the way, to my knowledge, the main goal of New Translation Bible was to make it more precise - Synodal Bible is criticized by many in that aspect - rather than to use more modern language. The question whether this goal was achieved is out of scope of this Stack and one can ask at Christianity SE.
You can compare both translations by yourself: here's the New one, here's the Synodal one.