It's not a matter of "conjunction" or "preposition", it's a matter of amphasis.
If somebody asks, e.g., "Who wants one?", you may decide to indicate that you do - in which case you might answer "I do!", or alternatively "Me!". But you would not answer "I!"
This is because, as well as being the form used as an "object" ("The ball hit me") and after a preposition ("Give it to me!"), "me" is also the emphatic form of the first person singular, the form used in the absence of a verb.
Thus, "better than me" is always to be preferred to "better than I" as the pronoun has no verb and is, therefore "emphatic", but "better than me am/is/are/whatever" is wrong, as the pronoun is being used as the subject of a verb.