The ends of my rim tape are about 1mm apart, but the spoke holes are covered. Is this ok, or should I pull it off and start with a fresh roll? Here's a picture.
This is usually ok. as long as it has strong enough adhesive to stay in place. It's better to overlap it, of course, but as long as the spoke holes are covered it should be ok.
A bigger issue is that usually, when the tape comes up short, it means that roll was designed for a 26" wheel, and is going on a 700c rim. If that is the case, make sure the tape is not too wide, because if it is, it prevents the bead of the tire from seating, by filling some of the space in the beadlock on the rim.
If it curls up on to the side walls of the rim at all, that is not a good option. There are different widths for 26" tape, so even if it is 26", it can still be the correct width.
You should be fine - the primary purpose of rim tape is to protect from the spoke holes. If you are at all concerned that there's some rough in that small gap, a small peice of duct tape to bridge the gap would probably suffice (when isn't duct tape the answer?).
I wouldn't waste a roll of tape for the sake of a single millimetre!