I do not think it will go upright by itself anymore. Opuntias are completely happy to grow in pretty much any orientation and position. They are notorious spreaders and space fillers and I would consider almost all 'messy' growers. The white things look like they might be 'air roots', which if they do find soil will start behaving like normal roots.
If you really need it to be upright, the only real option would be to prune it right at the base of the sideways pad, let that dry out for a few days and then put that in the soil with the new growth facing towards the window. You could combine the pruned part into the old pot, but you will want to stop watering for a week to two weeks to let that pruned section develop roots, which since it appears in the growing section might be relatively quickly.
