NYC Boy in LA for a few days
For Korean BBQ the board favorite seems to clearly be Park's.
955 S. Vermont Ave. (at San Marino Street)
Los Angeles (213) 380-1717
For Sushi you will close to two board favorites. Sushi Sushi and Sushi Zo. I am partial to Sushi Zo and the way to order there is to put yourself in the chef's hands with omakase.
326 1/2 Beverly Drive
Beverly Hills, CA 90212-4801
9824 National Blvd Unit C
Los Angeles, CA 90034
The drive from Beverly Hills to Little Tokyo for lunch may not be worth it -- plus there's parking to contend with. Not to dismiss Sushi Gen; just don't think it'll be worth your time and effort and you'll end up hating LA.
If you want sushi and price is a consideration, why not head west to Sawtelle, take a left and just north of Olympic, on Sawtelle in the Olympic Collection (upscale mini-mall) is Kiriko, which has a very good lunch omakase special for around $30 that includes soup and salad. The added bonus of being on Sawtelle is that you will see some really good options for your other eating times -- Orris is Franco Japanese, Chabuya is very good noodles, there is a Korean BBQ place I think, in the strip mall just north of the Olympic collection on Sawtelle, as well as a number of other very worthwhile, not too expensive places. I mean if I had to take one street from LA to the proverbial desert island and eat only at restaurants on that street forever, Sawtelle would probably be my choice.
Even closer to you, and also excellent, but pricier, is Sushi Sushi on Beverly Drive in Beverly Hills.
Absolutely agree with NAspy.
For your location and the benefit of not having to sit in traffic, the Sawtelle strip of restaurants is great to satisfy your Japanese food cravings.
My favorites on that row are Kiriko (sushi and cooked items), Hurry Curry (curry), Sawtelle Kitchen (yoshoku), Furaibo (izakaya), Orris (French-Japanese fusion), and Chabuya (ramen).
Also nearby, the best ramen in the area for my money is the Santouka ramen stall in the Mitsuwa supermarket food court on Centinela. Also love Musha in Santa Monica on Wilshire for izakaya in a slightly nicer atmosphere. Ramen-ya on Olympic is Korean-Japanese ramen so that might hit both your cravings.