Do the studios at A-STAY Studios have a kitchen?

Yes. Every studio has an open kitchen, enough to actually cook, not just reheat:

Ceramic hob with extractor fan
Microwave grill
Dishwasher
Fridge with freezer compartment
Coffee maker and kettle
Crockery, pots, pans and utensils
Dish soap, sponge and the rest of the boring bits