If you have a 4 hours of free time from 8am to 12pm in Taipei on a Monday morning, what would you do?
If it's not too rainy, this could be a wonderful time to visit the Taipei 101 Observatory. Should be less crowded, and can see some of the morning hubub of the city coming to life for the week ahead.
Alternatively, if you want something more active and want to go for a hike, I recommend Bishan Temple off the Neihu station MRT (I believe Red Bus 2 goes to the temple from there). It's a bit of a walk, but you can also access hiking trails from the temple up to Mount Zhongshan.
Mentioned in this answer:
One of my favorite places in Taipei is the Taipei Botanical Garden! It's a beautiful, quiet, relaxing place with so many interesting things to see.
Breakfast! The food is quite interesting (not talking McD or Starbucks here)
And best time to go National Palace Museum before the mainland Chinese tourists come in droves (not exaggerating, it was really crowded by midday).
Taipei has an absolutely wonderful zoo (Taipei Zoo). It opens at 9:00 a.m. Use public transit -- take the MRT Wenshan-Neihu Line and get off at Taipei Zoo Station.