The rule of thumb with overpainting (or over-finishing, since this applies to clear finishes also) is that as long as the surface is completely clean, dull (matt) and dust-free you should get a decent bond1.
Should I use a different sanding paper grit?
220 is a good choice. You could go a little coarser2 or a little finer and still get acceptable results, but 220 is a good mid-range grit for this type of job.
Is there an alternative to rubbing alcohol?
Just simply cleaning with hot soapy water may be enough to clean a surface sufficient to start the prep for refinishing, but it depends on what it is/might be contaminated with.
Waxes aren't soluble in soapy water (even if the water is hot) so a solvent of some kind would be needed if any furniture polish or other product containing wax was ever used over the existing paint. Mineral spirits (UK: white spirit, Oz: mineral turpentine) is a good choice for this purpose, plus is also ideal for removing greasy or oily residues without the minor risk of exposing the piece to water. And generally speaking no fully-cured paint is adversely affected by mineral spirits.
1 This does also apply to polyurethane, putting the lie to the conventional wisdom that you can't overcoat stuff originally finished in poly.
2 When painting you can sand more coarsely than when using a clear finish without it being a problem.