In order:
* LaJolla to border is 30min or less; you could leave your hotel and be sitting down to dinner somewhere in TJ in less than an hour (other then 3pm-5pm).
* time to get across going into Mexico is solely the amount of time to walk the space. I'd say 20min or less, depending on age & fitness.
* right along the border, it's TJ or nothing, until you get over to Mexicali (about 90-120min drive East). However, you could easily jump further South to 3+ locations:

1) ABC bus from TJ border costs $10 or less and takes you to Ensenada, a really nice coastal city, albeit much nicer other than 11am-6pm Tue-Sun when cruise ships are in. Really pretty harbor city, some killer seafood, and more laidback than Tijuana
2) Rosarito is a smaller tourist spot about 30min South of Mexico, with nice beaches, and very tourist oriented. Puerto Nuevo is where white people eat lobster.
3) if you had the time & the spirit, 90min from Tijuana is Valle de Guadalupe, Northern Baja Wine country. I've only been once myself, very recently, and was blown away by the landscape, the food, the wineries. A good deal pricier than TJ but freaking gorgeous.

If you've only got the 1 day/night, hit TJ, wander down Revu from Arch to 9th street, stopping anywhere for food & bev, or grab an Uber to Zona Gastronomica for high end eating, or Telefonica for probably the best food for the money you'll have in North America