Yes! In February 2022, Rumpke announced it will now accept disposable paper, plastic, and aluminum cups in its recycling programs. Styrofoam and polystyrene (#6) cups (e.g. solo cups) are not accepted.
As for the lids, plastic lids on plastic cups are accepted and should be reattached prior to placing in the recycling container. Plastic lids on paper cups are not accepted - the paper cup should be placed in the recycling without the lid and the lid should be discarded in the trash.
Straws and stoppers should also be discarded in the trash. This material is much too small to effectively run through the recycling facility.