“Can I get a sip of that?”

You're out on a walk with your girlfriend. It's hot out, and you're sweating and thirsty. You didn't bring any water, but your girlfriend did. She's drinking it, and you'd like to have some too. You say:

Can I get a sip of that?

Can I get (something)?

When you're asking for something small (in terms of value) you can ask in this way:

Mom, can I get some money for lunch?

Can I get your email address?

Actually, can I get plastic instead?

a sip of (a drink)

A "sip" is a small amount of a drink. It's less than one mouthful. 

When you're asking to drink someone else's drink, it's common to ask for "a sip". That's because you don't want the person to think that you're going to drink a lot of their drink!

If you want to talk about drinking a large amount of a drink all at once, use the word "gulp". It means a full, large mouthful of a drink.

To describe swallowing a medium amount of a liquid, call it "a mouthful".