There are several ways to get to the Roissy (CDG) airport from the Arc de Triomphe (usually called “Place de l'Étoile” or just “Étoile” by the locals, and sometimes known as “Charles de Gaulle” on transport maps — not to be confused with “Charles de Gaulle” refering to the airport!).
Air France runs coaches to the airport, and one of the lines goes from Étoile to CDG. That's by far the simplest way for you, but they might not start early enough. They're more expensive than the train or even than a night bus plus train, but a lot cheaper than a taxi. The first bus departs at 5:45 and the journey should take a little under 1 hour.
RATP (the local public transport company) also runs buses to CDG. The buses are only slightly more expensive than the train. The downside of RATP buses is that they leave from place de l'Opéra, a little over 3km away from Étoile. The upside is that the first bus leaves Opéra at 5:15; the journey takes about 1 hour.
RER B is a commuter train that runs through the center of the city and has a branch that terminates at CDG. You can find the full schedule on the RATP website (pick direction “Charles-de-Gaulle”, and “lundi à vendredi” for weekdays or “samedi, dimanche et jours fériés” for week-ends and public holidays). The first train arrives at the airport at 5:31. It departs from Gare du Nord at 4:53 (from the upper-level platforms, unlike later trains which go through the underground platforms). The first train from the central station Châtelet-Les-Halles departs at 5:26 and arrives at 6:05, after this there are trains every 8 minutes or better.
From Étoile, the most direct way to catch line B is to take RER line A to Châtelet-Les-Halles. You'll need to change platforms at Châtelet-Les-Halles: lines A and B share a platform but that would be the train in the wrong direction. The first train from Étoile leaves at 5:13 and arrives at Châtelet-Les-Halles at 5:19. Note that from 27 July to 07 August 2015, traffic on line A will be suspended around Étoile, so if you're traveling during this period, you'll need to take a bus to Auber then change for a train; these deviations are usually well sign-posted, but they add considerably to the journey time.
If RER B is fine but RER A isn't, you may want to take a taxi to Gare du Nord. It'll be a lot cheaper than going all the way to the airport, but somewhat slower.
Paris has a night bus network. You can use an ordinary ticket (“ticket t+”) for a trip inside Paris. If you go all the way to the airport, you'll need to stamp 4 tickets. If you change buses, you need to stamp a new ticket. Many lines are infrequent, so do check the schedule in advance and arrive at the stop early. Stops are clearly marked with the “N” logo but make sure you know where exactly to find your bus as Place de l'Étoile is very large.
- From Étoile to Gare du Nord, there is no direct bus. Check the schedule planner to see the possibilities.
- If you want a lift towards Opéra, take N53 to Saint-Lazare and walk about 1km.
- On the other hand, Châtelet-Les-Halles is easy to reach, because it's next to the center point of the night bus network. Take N11 or N24 and get off at Châtelet.
- Line N143 to Roissy airport leaves Gare de l'Est (only a few blocks away from Gare du Nord). If the earliest RER B is too late for you, N143 can get you there earlier; it runs every half hour and the journey time is an hour or less depending on which terminal you're going to.
If you decide to take a taxi for all or part of the journey, there are companies that let you book online. If you're going to pay by credit card, make sure to take a booking that guarantees the possibility — some taxi drivers can't or won't take cards. The price for the trip to Gare du Nord should be around 15€, to the airport might be something like 60€. Alternatively, you might want to book a car-with-a-driver-that-isn't-a-taxi (“VTC”), which offers a pre-arranged fare and almost always accepts payment cards. Unlike taxis, VTC fares are not regulated, and may end up being more or less expensive than a taxi.