The bus will also depend on what time of day. Anywhere from 7am-10am the bus is going to encounter heavy traffic on the freeway going into Taipei, so the airport MRT commuter train to SanChong and change to the orange line which would get you to Zhongshan Elementary School station avoiding traffic.

The 1841 bus is the 'commuter' version which goes via the cargo terminal and other airport facilities, then takes local roads to the freeway, so could be a very long ride especially in peak hour.

Another bus option is the 1840 (destination Songshan Airport, cost around NT125-ish) which goes express on the freeway from the airport to Taipei city. You could get off at Xingtian Temple (the first stop after the freeway) and a taxi to Aloft would be around NT100. If you're light on luggage you could walk to Xingtian Temple MRT and take it one station to Zhongshan Elementary School station.