the most reliable and affordable way to the Airport is the Taipei MRT. i am not sure however what time to they operate after new years

depends on what you would like to see. If you see taipei by night, then 5 to 6 PM is ok. if you want to see taipei by morning, around lunch time is ok. just make sure there are less fogs so that you may see the view clearly

Based on the timing of your layover in Taiwan, there is definitely sufficient time for you to travel to Taipei 101. From the Taoyuan International Airport, catch a 5-minute shuttle (green Ubus) to the High Speed Rail (HSR) station. Buses run every 10-15 minutes from 7:00 am. Take the High Speed Rail to Taipei. Travel time is between 20 to 25 minutes at TWD350 per adult on the standard return ticket. There is the earliest arriving train at Taipei Station at 7.30 am. From Taipei Station, change train on the red line toward Taipei 101 Station. The Taipei 101 is open between 9.00 am to 10,00 pm daily. Ticket is priced at TWD500 per adult and this can be purchased on level 5. While at the observatory between 9.00 am to 10.00 am, you will be able to witness the shadow of Taipei 101 cast against Elephant Mountain (象山, Xiangshan). If you intend to have breakfast up at the observatory, you will be disappointed. Unfortunately shops do not open as early as you will arrive at Taipei 101. Shops open at 11.00 am. You may have to look for simple breakfast at either the closest 7-Eleven or McDonalds. And because you have to be back at the Taoyuan International Airport by 12 noon, you wouldn't have time to experience the local fare. If only you had more time, I would have recommended for you to have a good brunch/lunch at Din Tai Fung located in the basement of Taipei 101. In fact, I would recommend an overnight layover in Taipei before you head to Bangkok. There is a lot to see and local food to savor. P.S.: Just so that you know, there are food choices available at the Taoyuan International Airport. These can be found in the transit areas. Value for money. Have a good time in Taipei and safe travels.