No train yet
Unfortunately, the airport that was over 40 years in the planning and construction STILL does not have the transit link functioning. While everything seems to be ready and test trains regularly run the route, it is not yet open. No date has been announced, but in typical Thai style, someone is doubtless waiting for an appropriatly auspicious date.
I suggest that you either take a taxi (inexpensive, especially by western standards) or have your hotel arrange for a car. The latter will be more expensive, but if you're a BKK newbie, will remove any doubt from where you may be going or how you will get there. Taxis are usually no problem, but you do occasionally get the bad apple.
Enjoy your stay in the City of Angels!
Cheers...Dan