ProleOnParole

OK. Let's see...

Option #1, assuming the Nangang HSR station opens as planned on July 1 and doesn't close on July 2 after some SNAFU:

Ubus 705, every 10 min, travel time 25 min, 30 TWD

Note: limited seating and luggage space, essentially like a city bus, wouldn't recommend doing it with more than one large bag per person. Buses for Taipei have luggage compartments and everyone gets a seat (or has to wait for the next bus).

HSR Taoyuan-Nangang, every 20-40 min, travel time 30-35 min, 190/200 TWD

You can book online for a discount of up to 35% (certain trains, 5-28 days in advance) but every ticket is for a specific train only: miss it and you have to pay for another one with no refund.

I did a trip like this last year (to Gaoxiong). I booked early allowing some leeway for flight delays, and ended up arriving at the station 2 hours ahead of the train I had the ticket for. The reservation can be changed at the counter but the lines were so long I paid for an upgrade to business just to skip them (there's a separate counter that can be used for business bookings, otherwise the upgrade is not worth it). I think it might have been a Friday afternoon though, so it's not inevitable. Taoyuan-Nangang in business is 400 TWD.

Note: there are HSR tourist passes allowing unlimited travel for a number of days, maybe this changes the calculus somehow if you already have one or plan to get it anyway.

Yet another note: HSR is not really high-speed north of Taoyuan and goes in a tunnel much of the way so don't choose this option just for the HSR experience. Not worth it.

Total time: 1h 15 min assuming no wait in line for the ticket
Total cost: 220 TWD

Option #2 is to take bus to Nangang Exhibition Ctr:

Guoguang 1843, every 1h, travel time 1h 20 min, 110 TWD

There are three problems with this bus: (1) it doesn't go very often, (2) it gets off the freeway early to stop in Neihu, and then has to get back on the freeway again, which unnecessarily adds another 15-20 min to travel time, (3) adding insult to injury, it then drops you off one MRT station away from the hotel, so you can either take the:

MRT Nangang Exhib Ctr-Nangang, every 3-4 min, travel time 2 min (+5 min to get in and out of the stations), 16/20 TWD

Or just walk for 15 min or 1 km (not recommended with luggage).

Total time: 2h
Total cost: 130 TWD

None of the above looks convenient so far. Personally I'd go for:

Option #3, take the most frequent and quickest bus, and transfer to MRT at the first opportunity:

Bus 1819, every 10-20 min, travel time 40-45 min (to Kulun St), 125 TWD

Let the bus continue to Taipei Main Station in traffic jams for another 10-15 min while you get off at Kulun St and walk across the street for less than 5 min or 150 m to MRT Yuanshan.

MRT Yuanshan-Nangang, every 3-4 min, travel time 30 min, 28/35 TWD (change at Taipei Main Stn for a blue line train towards Nangang Exhib Ctr)

With a lot of luggage, you could alternatively take a taxi from MRT Yuanshan, typically for ~ 150 TWD up to ~ 300 TWD for very remote destinations.

Total time: 1h 25 min
Total cost: 160 TWD

There are dozens of hotels that won't need so much gymnastics to get to as they're reached directly by one or another bus from the airport, so you can't really say this one is convenient for airport transfer. The hotels easiest to get to from the airport would be somewhere near MRT Minquan W Rd or (MRT) Xingtian Temple. Not that I would particularly recommend to stay in these areas (I wouldn't).

Also note: bus 1819 doesn't go the same way back to the airport, so the best place to get on is unfortunately the Taipei (Guoguang) Bus Station (MRT Taipei Main Stn, exit Z3).