Answer

The cup has not been immersed. True, in the past I have seen halakhic decisors who wrote that there is no prohibition in this if the cup is not yours, since the obligation to immerse it is a personal obligation on the owner. However, according to the overwhelming majority of halakhic decisors, there is a prohibition even with a cup that is not his. Still, it is not clear that you have an obligation to investigate whether he immersed it and whether this is a cup that requires immersion at all (after all, you are not drinking anything forbidden). And if this is a business, there is more room to be lenient, as is known, because the utensils are not for personal use. Especially if it would insult the host, there may perhaps be room to be lenient. But it is preferable to ask for a disposable cup.