If I open a bottle of beer and don't finish it (I don't drink cheap
beer!), is there a pressure cap or device that will keep it carbonated
and relativley fresh until opened again?

Don't finish a bottle of beer...hmm, interesting concept.

Anyway,

"Seal your bottles for longer lasting taste with this amazing designer
bottle stopper. Use this bottle stopper to seal your wine, beer, or
liquor bottles to lock the freshness in for weeks. You no longer have
to open the bottle and finish it. You can now enjoy one glass at a
time. Your fifth glass will be just as refreshing and enjoyable as
your first. Simply place the cork bottle stopper in the bottle, push
down the handle and the o-ring expands sealing the bottle airtight.
This is a perfect gift for any wine or beer lover. These plastic
stoppers come in a set of 3."

http://www.stacksandstacks.com/html/11182_bottle-stopper-set-of-3.htm

I hope this helps.

Have a good day.

Short of a stopper which creates a vacuum, such as VacuVin,
which is used to preserve wines, you can just use a cork
from an empty wine bottle. Some loss of carbonation will
occur, but if you drink the beer within 24 hours, you may
find it acceptable.

If not, use something like VacuVin:
http://www.vacuvin.nl/wining_winesaver.html

I rather think a vacuum stopper like the vacuVin would draw out the
carbonation in beer, making it even "flatter".

All stoppers that stop the escape of carbonation will be equally
effective, such as the one suggested by Thither.  There are many
other.  Keeping the opened bottle in the fridge overnight will also
reduce the loss of carbonation  - even if you don't drink it that
cold.

Despite these measures, the emptier the bottle is, the more
carbonation will be lost, since the greater volume of air in the
bottle will absorb (?) more CO2.

Obviously, it would also be best always to cork the bottle after each glass poured.

Personally, I agree with Thither's opening line, but respects to those
who choose not to finish an opened bottle of beer.

Cheers, Prost, Myoarin

You would have to be joking. What sort of a beer drinker are you?
Genuine beer drinkers NEVER recap an open bottle/ stubbie/can. They
FINISH IT. and then if you have decided that you have had
enough...don't open another.