Stripping is the way to go if you really want the job to last. Frankly, the time & effort to properly abrade the existing paint w/sandpaper to ensure adhesion of your new paint makes stripping the better choice, IMO. Chemical deglossing as mentioned above is an easy alternative, though.

As an alternative to blasting the rust look into commonly available chemical rust removers like Naval Jelly. Again, follow label directions. A steel wire wheel on a power drill can do the job too.

It's true that "rust never sleeps" but with a little extra care in even a DIY job you can keep it at bay for a surprisingly long time.

Again, bear in mind the OP called the bike a "beater" and is presumably seeking a cheap DIY solution. So powdercoating or other professional finishes, for all their undeniable virtues, are off the table.