Might need to clean out the BEAD SEAT area of the rim. So many old bikes have rotted rubber in the crevice that should hold the tire bead in. It doesn't take much to have this happen. If that does not do it, look at the bead on the tire and see if that is the issue. The bead needs to lock in, if it doesn't, it will roll off like you state.
The other thing to do is fill up the tire and tube with about 20 PSI and try and hand seat the bead on the rim. Make sure that it is even all the way around on BOTH sides. Then add more air and check again and again until you have the proper pressure.