taxi tour around Incheon Airport to Yonggungsa Temple
 

We have a 5 hour layover at Incheon airport arriving at 5:30 am in April. We were wondering if we could get a taxi driver at the airport to take us around to see Yonggungsa Temple (read that it is lit up with lanterns on the steps at night)and then maybe stop in at a Korean restaurant for breakfast. We know we won't have a lot of time but thought it would be nice to get out of the airport before the long flight back to the US. Has anyone done this and know approximate cost for a private taxi for an hour or two? The airport transit tours do not start until 8 or 9 am which would be too late for us.

would the temple be open that early?

isn't yonggungsa temple located in busan? if so, then you won't be able to visit it b/c busan is several hours away from incheon.

I was confused by the temple names. I saw that there is a one hour transit tour to Younggungsa Temple which is supposed to be 20 minutes from Incheon airport.( There is also a similar named temple which has the lanterns that is near Busan). I just realized the temple does not open until 9 am but maybe we can drive by it. Here is the transit tour description:
Enjoy a one hour Younggungsa Temple tour located at the foot of Mt. Baegunsan nearby the airport. Built during the Silla Dynasty in 670AD this tour allows an insight into Korean history while you await your onward flight. The free one hour temple tour operates three times per day departing Incheon International Airport at 9am 11am and 3pm.

Where did you find the info on the tour? Thanks