My attitude is similar to what you and @olm wrote above. We put together a small backpack (two, actually) a couple of months ago in the run-up to the U.S. election (mostly at my girlfriend’s insistence - I’m not sure personally how useful the bags would be in the event of anything happening, but it’s definitely helped with her worrying/complaining about it, and ergo my stress levels).
The bags contain:
- passports
- some irreplaceable documents
- some USD (I should probably get more, or some JPY/CNY). I tend to keep a fair amount of TWD on me anyway, since I almost entirely use cash and don’t like to go to the ATM too often.
- a first aid kit and basic medicines
- a few masks (N95 and surgical)
- a couple of days of food (tinned tuna/fruit, instant noodles, cereal bars, etc.). I may actually have eaten the instant noodles and cereal bars already - I should replace those.
- a small amount of camping stuff (tiny camping stove/pans/gas cylinder/lighters, multi-tool, knives, torches, emergency blankets)
- some bottled water (probably not enough, tbh - if there was any kind of warning I’d probably throw the above small backpack in a bigger backpack with one or two 6-liter bottles of water we have sitting around in case of typhoons etc.)
I did think about a physical map, but I also know that if we have to for some reason make it to Nantou or whatever through the mountains we’re probably screwed anyway. 
I did also consider some of the things that others have mentioned (better face masks, water purification tablets, pepper spray, etc.), but figured those might be entering paranoia territory.
I’m not sure about the best immediate (hours/days) strategy. My assumption was that Taoyuan airport would be put out of service pretty rapidly in the event of an invasion and, even if it wasn’t, getting commercial flights out would quickly become close to impossible and cost-prohibitive (not to mention a lot of Taiwanese suddenly realizing they’re actually American or Canadian and wanting to go home). I’d probably first want to get to some shelter/less populated area (riverside park/MRT station/emergency shelter - is there a list/map of those somewhere?) and then away from Taipei, but I’m not sure how feasible any of those things would be.