Hahaha, for real. This is exactly what happens when you don't heed advice... I didn't do my homework... So I rushed to post this as a reminder to everyone: absolutely, positively use public transport!
I think shorts are fine; just bring a jacket. The key is to bring flip-flops or buy a pair here โ they're essential for rain and comfortable even when it's not raining. For seven days, I think 13,000 TWD should be enough unless you're planning on expensive activities or restaurants.
The Sicao Green Tunnel, Chimei Museum, watching the sunset at Yuguang Island, and exploring the creative shops and historical sites on Shennong Street โ these are all definitely worth visiting!