Q:

Do Americans drop the 'a' as in 'that's good time'?

Do Americans tend to drop the indefinite article in spoken English? I'm under the impression that they do. 
I've just watched a movie in which two guys were chatting about horse racing. One of them mentioned a certain horse's performance and the other replied with "that's good time". 

A:

No. There are a number of different idiomatic uses of time (most of them with good time) that do not need an article. They usually have a meaning of either promptness or speed.
In the context of a horse that's good time means that the horse has run the course at a good speed - i.e. has taken a good (short) time to run the course.
Other related phrases are:

in good time = "early enough"
make good time = (arrive) earlier than might have been expected.

