The "Flying Dog" bus line, which has a stop right in front of the Westin, might stop running a bit earlier. But Airbus has a stop on Nanking East Road, which is drop-off only. That's less than 75ft. away from the Westin Taipei if you don't have heavy luggage.
If you do, get off at the next stop and call a cab. No cab will pick you up only to take you 75ft. away. It'll be only $150NT. Added to the bus fare of $140NT, that's less than $300NT.