What you mean by cheaper? Air ticket/ hotel
You travel in low season sure cheaper but you have to worry of rain and typhoon, high season will be more expensive unless you book during promo.
HSR only 45mins, you want zhun zhun, buy hsr ticket on the spot then board the coming train.
If have express train even faster since lesser stop.
What I'm saying is you take bus or Hsr not much of a diff because need to take mrt or bus which is about 15-20mins already and take hsr again about 40mins excluding the waiting time which is more or less the same. The only diff is pay 200-300ntd extra for maybe 15-30mins faster?