"How many meanings are there for this word?" should be acceptable too, right? Or would that need בשביל ?
Yes of course your translation should be acceptable, it's perfectly normal English. בשביל suggests you are doing something "for" somebody's sake or it's a present "for" them; you wouldn't use it here.
Kama mashma'uyot yesh la'milah ha'zot?
How about "This word has how many meanings?"
זאת שאלה טובה לדעת בעברית!