If your subscription is active, and Lightroom on your current machine has fully synced with Lightroom in the cloud, you should be able to go to a new computer, install Lightroom, log in with your Adobe ID, and you will be in business. This is the value proposition of Lightroom (the cloud version).
That said, if you have custom presets (or profiles), it would not hurt to have a backup copy of your presets exported and saved, so that you can re-import them on the new machine.
Wow, OK Lightroom 4 from 2012.... That's an entirely different problem! I thought you meant Lightroom v4, which is the current cloud-based subscription version. There was no syncing in the old Lightroom 4 (the current version of what used to be that program is now called, unfortunately, "Lightroom Classic.")
Wobert's link above is probably the best place to start, although keep in mind that version is no longer supported, so any anomalies you are on your own (with whatever help we can provide here, of course, but no help from Adobe).