We are staying in Luxembourg for one night before an early flight the next day (6:30 a.m.). Will it be possible to get a means of transportation to the airport from inside the city that early in the morning or will we need to stay at one of the airport hotels?
the hotel has a courtesy car that you can arrange for in advance to take you to the airport or train station. You can email there concierge to take care of it in advance
If you stay at Le Royal, you can order a free transfer through the Concierge.
I'm certain the hotel will order a taxi for you.
The airport is only 5 miles from the city centre. There are many taxi services in Luxembourg so this won't be a problem. Ask the hotel to arrange the taxi for you if not then there is always Uber ;) Have a safe trip.
Yes I would recommend to take a taxi. Webtaxi is the name of the company with the best prices and reliable. I booked them for 01:15 am and they showed up right on time.... the reception can book for you i'm sure. Webtaxi can also tell you beforehand what the fare cost will be.
Get an answer from Le Royal Hotels & Resorts - Luxembourg staff and past guests.