Approximately how long does the average person spend playing one
session of a modern coin operated arcade game?

Coin Operated = An arcade game in a cabinet that requires you to
insert quarters, bills, or a time card (Dave & Busters or Gameworsk)
to play.
Modern = Preferably 2005 or newer
One Session = Without continuing by inserting more money

I would say there is no set time. It depends on a lot of variable
factors, the players skill, the game they are playing, the difficulty
of the arcade unit (the dip switch set up)

Because of the dip in popularity of Arcade gaming. I would say that
the average time of one-shot play has probably risen in recent years.
To entice players to stay in the Arcade and play.

From personal experience - 

Outrun 2 - This arcade is a racing game which runs over 5 stages.
Usually on one play an average player can reach stage 4 to 5.
(Choosing the easy route) - the average time for each stage is about 1
minute. So therefore an approximate play time would be 4 to 5 minutes.

The upshot is that it is very unusual for an arcade game to last more
than 10 minutes without additional coins.

My suggestion would be to extend further research by looking at an
emulation website such as MAME http://www.mame.net/ - although these
are not 2005 arcades, you will find examples of average play length by
playing them yourself.

The problem could be looked at from a different angle:  how much does
an arcade game take in  - is set to take in -  during continuous play?