The first one is not correct because "lipstick" is an uncountable noun and is not preceded by "a: indefinite article".
To correct it, we could simply leave out the article: Do you put on lipstick?
However, this simple correction is not sufficient because we don't really know what one means to say with this sentence. Also, consideriing the sentence as it is, without context, I think most people would choose the verb "use" instead of "put": Do you use lipstick? The verb "put", however, is possible.
Here are some other possibilities, depending on what one means to really say:
Do you use lipstick every day?
Did you put some lipstick on?
Are you going to put on any lipstick?
Are you going to use lipstick?