"Mail" is not a countable noun. You cannot say "one mail" or "seven mails." So, "doesn't receive much mail" is correct. If you used "Letter" instead, that is countable, so "doesn't receive many letters" would be the correct phrase. "Email" can be both - "doesn't receive much email" and "doesn't receive many emails" are both acceptable.

The sentence is correct, and for most native English speakers, the use of "mails" or "many" would sound strange if used in this sentence. This is mostly something you pick up from speaking English for a while, but I'll try to explain it as best as I can. First, "mails" isn't a word, so you don't need to worry about that one. "Many" is used to describe the plural of a noun. For example, I saw many cats today, cats being the plural of cat. It sounds strange to most English speakers if you said, I saw many cat today. Thus, "many" is used to describe plural nouns. However, there isn't really a good plural word for mail, so "many" is simply not used at all in this context. Hope this helps ^_^