As I recall, citizens use different lanes from non-citizens in Taiwan. Speeding inspection has nothing to do with cabin of services.

Speedy services only apply to diplomats and certain businessmen that meet the criteria. For foreign businesspeople with need for speedy immigration clearance, there are some options:

If you are referring to"Fast-track Passport Control for Frequent Visitors", you should look into applying for a Speedy Immigration Inspection Certificate here. Once approved, you may use "Foreign Frequent Visitors" lanes upon arrival/departure at the airport.

Starting in 2011, there's also a paid service to use Huan-Yu VIP Terminal (環宇商務中心) at TPE so visitors may enjoy "expedited check-in and immigration clearance" (商務禮遇), among other services. I believe the rate is NT$8000 per person but you should call them directly for more details.

As for your plan to visit the city (presuming Taipei City?), you didn't say what day you are arriving but the early dawn happens to be mini-rush hours since quite some folks choose those arriving flights so they may start their (working) day right off the planes.

Besides, your arrival/departure is really too short for a meaningful visit. I am not even sure you would get 2 full hours in between if the traffic is not cooperating. Of course you don't get to see any night market during those morning hours. Some traditional markets, maybe, but I don't know if that's your cup of tea. Temples, yes, that's one possibility but it depends on where you end up visiting & what temples interest you.

If your departure flight were in the late afternoon, I may be able to suggest something. Meanwhile, you may check out what's been said in the other thread when someone had 12 hours layover in TPE. Good luck and hope you'll enjoy Taipei.