Can you explain why something of this modest size needs to be glued up?
If the liquids to be held are wine, red wine will stain the wood. With a dark wood, like walnut, it will probably not be very noticeable, but if it's maple, the owner will have a red colored vessel.
Poly doesn't hold up to liquids very well. It will shortly start to peel up.
One way to deal with both the liquids and the potential staining is to coat the inside of the cup with epoxy. I make fishing rods and have lots of practice with epoxy rod finish, which is fairly thin. You can do the same, and have it cure much faster, by using regular 30 minute epoxy and warm the A and B jars of epoxy in a water bath (or even 10-15 seconds in the microwave without ill effect on the epoxy). You have to slowly twirl the cup as the epoxy cures in order to get a thin, uniform coating. You could do the same thing with slow curing CA glue, but there are questions about whether it turns out food safe.
If these are not beverage containers, it's possible to size them to fit a glass tube or cup. Then the liquid is contained in the glass and the wood is not in contact with the fluid.