Actually, with reference to this particular example, the word
since would generally only be used in a negative sentence with a perfect-tense verb, such as "I haven't met him since Monday." (Also, in spoken American English, we would almost always say, "I haven't
seen him since Monday," not
met.)
As for your question,
since (and its counterpart
for) have different uses than prepositions of time (
on,
in,
at).
Since and
for are typically (but not always) used with perfect tenses, such as present perfect and past perfect (as well as their continuous forms).
You are correct to think that the word
since is used with a specific point in time (such as a day or date), while
for is used with a period of time, such as "three days" or "a month." However, neither of these words are interchangeable with prepositions of time.