As a Brit, I've always thought to "gee things up" (often followed by "a bit") was a relatively well-known Americanism - probably because I assume most figurative usages relating to horses come from the great American cowboy tradition.
In my usage, to "gee things up" always means to get things moving (normally figuratively, as of say action to deal with some problem). But I also hear/use "gee him up" with the sense of to tease or provoke, similar to rattle his cage or wind him up.
But checking gee things up in Google Books I found only 15 results, and to my admittedly untrained eye, they seem to be mostly British. Is the expression (and the tease/annoy meaning) familiar to Americans?