I got my PCR test at Shuangho Hospital （雙和醫院）. The testing process took about 1.5 hrs. Most of it was just waiting in line.
The outdoor testing site is right by the entrance to the Emergency Department（急診大門）. To reach it, turn left from the main entrance (where all the taxis are), turn right alongside the building, and then go straight. You should see a line or at least a group of hospital workers outside the Emergency Department entrance. See this map for more info.
I arrived at 8:30 am and was 21st in line. That day there were about 50 people total. I think they will accept anyone who is standing in line before 10 am.
Around 9 am someone will give you a sticker that indicates your position in line. They will also check your forms and ID and give you another form which you need to fill out.
Sometime later, someone else will come around and collect your documents and IDs. You will need proof of travel, your passport and any ARC IDs, photocopies of those IDs, and the self-expense COVID testing application form.
After a while they will return your IDs to you. When you reach the main entrance, you will be called into a tent area to another line. Once your name is called, you go into the testing room where they take a nasal swab. They give you a receipt afterwards.
You then take that receipt to the cashier inside the Emergency Department and pay 7000 TWD. As is common in Taiwan, my foreign credit card did not work so I paid cash.
After paying they will give you another receipt. Keep this to pick up your test results later.
You can pickup your test results at the main cashier inside the main hospital building. This is to the right from the main entrance.
You can only pick up your results after 7:00 pm. I went at 6:59 pm and they would not give me my results, so I waited a minute, took another ticket, and went back to the cashier. You will need to give them your passport and the receipt.