Which one sounds the best?

Examples 3 and 4 are just acceptable alternative ways of saying 1 and 2 respectively. Idiomatically, there is no difference. 3 & 4 seem slightly less polite to me as a native British English speaker, but perhaps not in other dialects.

The main difference is in the use of a plural "syrups" over the singular "syrup".

Syrup, like most liquids, is non-countable. If you had a full bottle of syrup and then divided it between two containers, you still have "some syrup". Likewise, if you were in a supermarket and you wanted to know where the bottled water was you would likely ask "where is the water?" even though there will be many bottles, different brands etc, it is generally accepted that there is only one kind of water - good old H2O. You wouldn't ask "where are the waters?" - although you can refer to the containers in the plural by asking "do you have bottles of water?"

With cough syrups it is slightly different. A cough syrup is a formulation, and so the different brands will be different kinds of syrup. Therefore, it isn't wrong to ask if someone has "cough syrups", in anticipation of them having more than one kind available. It is equally acceptable to ask if they have "cough syrup" in singular, because you likely only want one kind and the non-countable noun covers that.

The most idiomatic of your 4 options, to me at least, would be your first option "do you have any cough syrup?". Although if you are still unsure and want to eliminate any question over this, you could say "I'm looking for some cough syrup", which indicates your intent to purchase "some" and leaves it open for the respondent to tell you what is available to you.