Bakurocho to Narita doable?

I currently stay at a hotel near Bakurocho (and Asakusabashi) and saw trains bound for Narita AP. I suppose it would not matter if I took a regular train or a AP train express from Tokyo, Ueno, Shinjuku etc.. Is it doable? Considering having two big rollercarts maybe it would be better taking the express? If the train is all full I mean.

The trains in direction of Narita Airport will be very crowded in the late afternoons and early evenings of regular weekdays. During these times, it could be uncomfortable - although not impossible - to use the trains with large luggage.

Also, during the morning rush (around 8am to 9am) on regular weekdays, the trains could be quite crowded, although the main rush is in the opposite direction.

During all other times, the trains should not be overly crowded, but you may not necessarily be able to sit for the entire journey.

It's JR Sobu Line (Rapid Service)
http://www.japan-guide.com/e/e2027.html

Alright. The plane leaves at 11.20 (or 11.40). Aiming to be there at 9.00 (est). So I ought to leave around 7. Actually I was thinking of making a quick visit to the temple at Narita but perhaps it is too much of a rush.

Anyway, maybe it is better off taking the Narita Express from Tokyo instead?

Actually I was thinking of making a quick visit to the temple at Narita but perhaps it is too much of a rush.

You should calculate about one 45-60 minutes for that side trip, assumign a fast pace:
http://www.japan-guide.com/e/e6401.html

Anyway, maybe it is better off taking the Narita Express from Tokyo instead?

If you don't have a Suica & NEX package and wish to save some money, taking the Sobu Line would be recommended. It is also recommended for being direct, but there is the unknown factor of whether you will be able to sit or not. It ultimately comes down to personal preference.