1% Hydrocortisone is a pretty low concentration and if this is a one time thing then he likely didn't ingest that much.
Hydrocortisone is also a short acting steroid, so that means any effects that we do see will be short lived.
Possible side effects may be increased water consumption and subsequently increased urination. He may have a better than usual appetite too. But that doesn't mean he needs to eat more.
The cream or petroleum base in this cream, and sometimes steroids (although not usually at a low dose), can cause mild stomach upset (thus my question about drooling or lip smacking). Doxycycline can also upset the stomach too, so he may be more likely to get a little nauseous. Sometimes we also see slightly loose stools from the lubricant effect of an ointment or cream. But that should be self limiting. If that happens feed him a bland diet for a few days and he should come along.
A homemade bland diet consists of 1/3 boiled, lean hamburger (or boiled, white, skinless chicken) and 2/3 boiled white rice. Give small meals several times a day. Feed the bland diet for several days, then once his stools normalize start mixing in his regular diet and slowly convert him back.
High doses of steroids or repeated exposure can be immunosuppressive, which of course would be of concern in a dog with Lyme disease. But a one time low exposure isn't likely to be a big deal.
Sometimes we see increased panting with steroids, but I would not expect that at this dose.
Odds are he will remain perfectly fine with a one time low dose. We just need to make sure this doesn't happen again.
Please let me know if you have any further questions.