The great outdoors.
It's becoming ever more common to see tables with umbrellas perched outside around Taipei, but it has not always been as easy to find this type of outdoor ambiance. Smack in the middle of the bright, flashing, Asia-pop lights of Ximending sits the Red House Theater 西門紅樓, a historic monument which was built in 1908 during Japanese occupation. This charming red-brick relic of the past still has performances on the second floor. On the ground floor, you will find a cafe and gift shop--which is a good place to find kitsch Taiwan-themed souvenirs.
The courtyard surrounding the Red House Theater is the real attraction for the locals. Here you will find an astonishing number of tables and umbrellas clustered together in the outdoor courtyard area. The tables are attached to various establishments lining the side of the courtyard opposite the theater. It can often be difficult to distinguish one bar from the next, but if you pay attention to the general shape and style of the chairs, you can sort of guess which ones represent each bar. Choose your chairs wisely, for this may determine your drink selection for the night. Most offer a wide selection of upscale cocktails. If you are on a budget, check out the Thai restaurant closest to the entrance of the theater. It has the cheapest beer in the courtyard, at 50NT/ bottle for Singha or Taiwan beer, and the Thai food isn't too bad either. If the Thai restaurant's tables are full, they will deliver food to some of the neighboring bar tables, but unfortunately not the cheap beer.
A few practical notes: there is a public access restroom close to the Thai place, though it seems to inconveniently close in the early evening. It can be challenging to find an alternative, as not all of the bars have public facilities. Plan your drinking well.
西門紅樓 Ximen Red House Theater
Metro: Ximen Exit 1