Jaunted

Royal Claytons
Over by City Hall there's the self-styled British pub Royal Claytons. Their "Afternoon Delight" menu, available from 4pm-7pm, has $4 draft pints and $9 pizzas. They have lots of British-looking paintings on the walls and have "become a cool, hipster, hang out for all the nightlife goers." You can be confident that that's true because it's written prominently on their webpage.

Magnolia
The Hollywood-based Magnolia just got done with the grand opening of their new downtown outpost. Even better, they just got done with the grand opening of their new downtown outpost's 4pm-7pm happy hour. The $6 appetizers tend toward the pointedly bourgeois, being a mixture of fried goat cheese, calamari, and bruschetta. The food is matched by $7 cocktails and $5 wine, plus $4 domestic beers for the unrefined.

Ciudad
Upscale Cuban restaurant Ciudad is also willing to tolerate plebes during their 4pm-7pm happy hour (4pm-9pm on Sundays). Or as they describe it, you're welcome to immerse yourself in their "cosmopolitan vibe while sipping a minty mojito." If you're into discounted gourmet tacos and $4.50 mojitos. The mojitos, by the by, are rightly considered some of the best in LA.

Weiland Breweries
If you're looking for something a little less urbane - which is to say, something a little more fried - there are the two Weiland Breweries in downtown. Both have bar food-heavy happy hours from 3pm-7pm and 10pm-2am. The main one, open 7 days a week, is in Little Tokyo. The Underground one is in the center of the financial district and is only open on the weekdays. Getting there after dark requires traversing a darkened street, an abandoned escalator, and what very much appears to be a post-apocalyptic zombie-infested underground shopping mall. But hey, late night happy hour.