What does "sipping spirit" mean?

You are correct though I never used that expression cause I usually say I am sipping a gin and tonic or rum and coke or a black russian(my favorite cocktail)

Sipping spirit to me is any drink containing alcohol like whisky or vodka etc and taking small mouthfuls in a slow relaxed way. You can sip any drink, coca cola, coffee, tea, hot chocolate, etc and it means to drink small amounts slowly. 

In New Zealand, I refer to spirits as high alcohol and more expensive than wine or beer.

What does "Sipping a spirit" mean?

The answer to the question is no.   To sip is to drink very slowly

What does it mean when you say "a sipping spirit"?

you are right but this is not common usage.

I am a bartender.  A much more common way to phrase this would be:

Bartender: What can I get you?

You: I want something to sip on.  Can you recommend a nice whiskey?

or something like that.