I took shuttle bus from Taipei airport to 台中 (2-3 hrs) and then rented car to 埔里 (1hr +) last year.
Car rental in 埔里/Ching Jing
+886933179441
埔里黃包車家族 小楊
The schedule is feasible, yet a bit rush. You may visit 瑞士花園 late morning and then 青青草原 in the afternoon (but make sure there is the "sheep show" as it may not perform every day except holidays. Go to the hot spring at night. Leave Ching Jing early next day and go straight to 集集小鎮.