I would service it (clean, lube, adjust), replace the tires, tubes, rim tape, bar foam (I think it feels creepy) and brake pads and ride it just as it is. I road a 1983 12-speed Fuji Supreme, which looks a lot like that bike, for an entire summer a couple of years ago. I did a few group rides and never once had a problem keeping up, often to the chagrin of the people who had far newer, more expensive bikes.
If I had a 105 Group laying around I would be more inclined to look for a high-end bare frame to build it on and keep this Univega as it is for a backup/rainy day ride after putting some fenders on it.