how to get to old quebec area from new quebec?

I am traveling in Quebec city in January.
My hotel is Delta quebec hotel in the new quebec and I wonder how to get to old quebec area from the hotel.
I think I should take a bus because of the harsh weather.
Is there any bus that I can take from near the hotel to Old quebec ?
if it is not, how much will it cost by taxi?

Also I wonder how much it will cost from the hotel to the international airport by taxi.
I heard there is no way to the airport from the hotel area except taking a taxi, is it true?

Quebecers, help me !!

The Réseau de transport de la capitale (RTC) provides bus service in Québec city. Use the trip planner to find your route. The Delta Québec is at 690 boul René-Lévesque E (near Avenue Honoré-Mercier) – zoom in and find it on the map and you can use that as your starting point.

Taxi from downtown to the airport costs $32.50. RTC bus route 78 runs from Terminus Les Saules but there are only a couple of trips a day (and you would have to take a couple of buses from downtown to get there).