I've had a ganglion cyst on my foot aspirated 2 days ago. The tendons around the site are sore. Can I expect them to settle down soon?

Yes. It will improve over the next 3-5 days. Taking Motrin will help with inflammation.
Yes they supposed to. If the injection done by your specialist usually we mix two medication together, novocain (procaine) and steroid, when the novocain (procaine) wear off it hearts tell the effect of the steroid kick in in about 24 hours you should start feeling better.