No big deal. Do try to visit Narita after you drop your things at the NRT Hilton. Narita is a pleasant enough town that a lot of visitors shoot straight past on their way to the big city. If you make the effort to go there, take a stroll and have some dinner, you might be able to sleep past 4.00am the following morning.

If you're leaving early in the morning, it's possible to go either way. If you're going via town rather than via the airport, you should note a few things:

(1) SHUTTLE BUSES
Shuttle buses from NRT Hilton to town are less frequent than those to the airport. [TIMETABLE]. However, you only need one bus, so if there's one that suits your needs ...

(2) KEISEI TRAINS
The shuttle bus drop off point in town is in front of Keisei Narita station. From there, you can catch a Keisei "morning liner" or a "city liner" towards Tokyo. [TIMETABLE]

Since you'll be heading into Tokyo during the rush hour, and you probably have luggage, I'd suggest taking the Keisei train to Ueno and then getting in a cab. That way, you should have a seat all the way. Simples.

A more adventurous approach (and my preference) would be to take the Keisei train to a place called Aoto, where you can transfer (just across the platform IIRC) to a Keisei Oshiage Rapid train which is a through service that takes you on the Asakusa line subway tracks all the way to Shimbashi subway station - walking distance to the Conrad. Given the need to deal with train transfers, walking in unfamiliar streets etc. you might want to save that option for a future trip.

(Another option is to ride the Keisei train to Nippori and then get a Yamanote line train to Shimbashi, but that's a brutal ride in rush hour, so I don't advise it).

Keisei isn't part of the JR network, so JR passes aren't valid on Keisei trains.

(3) N'EX
Alternatively, you can walk through the Keisei station to the JR Station. Prior to 09.05am, N'EX trains stop at JR Narita Station. [TIMETABLE] So you can take one of those early N'EX trains to Tokyo Station. From Tokyo Station, I suggest you get a cab to the Conrad.

However, I don't think you can pick up the helpful N'EX + SUICA combo ticket at JR Narita Station. That is a nice money saver and I think it's only available at the airport. So if you want N'EX, I'd suggest going from your hotel to the airport and taking it from there.

Same story if you're planning to exchange a voucher for a JR pass - You need to do that at a designated station. You might as well go to the airport and do it there.