Its not possible to be both within walking distance to Ximenting *AND* Shilin night market. Unless you're a verocious walker. They're 5 MRT stations apart (i think). its like walking from pasir ris MRT to bedok MRT. heh heh... not impossible, but not what you might want to do.
Just in case, taxi fare within taipei itself is quite economical for 2 and above pple. No need to try to take bus if no MRT is available, just flag down a taxi.
*but*, taxi from airport to taipei is expensive. its about 1hr ride by taxi. the main reason is tht taxi needs special permit to operate to the airport. the fare is about NT$4000 i think. which is about $200. can try to bargain though I've nvr took cab before. They do have "private" taxi if you know who to call. it cost NT$2000 usually. "Private" as in illegal and they drive personal cars. IMHO, much more comfy than those legal taxi. (taiwanese are very entreprenarial). They'll offer u all sorts of service to take you around taipei & nearby. Very convenient if u intend on going outside of taipei to places like yeliu, shimeng dam, little world etc.
my usual ride from airport is using these private car "taxi", cost me NT$1200, but its to Hsinchu.
In taipei airport, there are buses to taipei. look for feigou (flying dog) operators. The bus hv dalmathians painted all over it. it'll take you to taipei, but not taipei MRT. just drop when you reach inside taipei and flag a cab to take you to your hotel. Depending on your drop site and hotel location, if u drop "accurately", taxi fare should not cost u more than NT$300, ie $15. I think the bus ride itself is NT$400, ie $20, though I'm not too certain on this. maybe cheaper.