Alex---the Lion's House building fron Venice Red House (name of the agency) is actually not at all close to Fondamente Nuova. I will tell you exactly where it is , if you have a map of Venice (or Google satellite view). find the Rialto Bridge (on a major bend in the Grand Danal) and follow the line doen and to the right (east) to find Campo San Bartolomio. Then let you eyes move due east over the red rooftops and across a small canal (Rio Della Fava). You may or may not recognize the little bridge over this rio on satellite view--itis hard to make out. Go just a bit further east, and you will see a small campo, named Campo San Lio. The Lion's House, comprising four apartments in all, is on this square---it is the building forming the south side of the square. At times we had a string quartet playing Vivaldi in the square.
This location is very central for walking all over Venice, which we did. We quickly discovered good routes to the places we wanted to go, such as the public pool in Cannaregio where my husband swam almost daily. It is a 5-minute walk to San Marco (if you don't make any wrong turns!). Thirty minutes to the train station. Fifteen more or less to the Accademia Bridge to visit Dorsoduro. And one of the best gelato shops in Venice (Suso) is a few steps away. Maybe in winter that is not relevant. . .
If you would like to be close to Fondamente Nuova, I recall seeing a nice 2-bdr. apartment close to Campo San Giovanni e Paolo on Airbnb. But I did not save the information.
As far as neighborhoods, I would characterize the northern parts of , on the edge of Venice, as less interesting---wider calles ( good for walking if you are in a hurry) but few shops and restaurants. And fewer tourists.
The next time we visited we stayed in San Polo, a bit north of the Rialto market. San Polo, with its narrow twisting streets, is where it is easy to get "lost in Venice"!
Our first visit, on our honeymoon 15 years ago, we stayed in Dorsoduro, which is many people's favorite, and may be mine too. I will look for something there next time. The Punta del Dogana---the tip of the peninsula that forms Dorsoduro, is a favorite viewpoint. We would walk out there every day or two, just because. I have read it is one of the beat viewpoints for the New Years fireworks, but I expect it would be very crowded.
There really is no bad location in Venice, as the whole island is very safe and traffic-free ( no noise). You just need to decide what you want, in terms of proximity to (or distance from) San Marco, Rialto Bridge, and the major sights. And amenities. For me, the rooftop terrace is #1 in requirements, so that makes it easy to filter all the available apartments. But in winter, you won't get much use from it, apart from the fireworks viewing if you want to do that apart from the crowds. (There is much to recommend both choices---private with your family, or joining all the others in a public place which could be lots of fun, or maybe too crowded to be enjoyable).
You can drive yourself crazy looking at apartments in Venicethere are so many. We have rented from Venice Red House and Views on Venice, and I would recommend both. Airbnb and sites like Flipkey, VRBO, and Homeaway have lots of listings for owner direct. I would be happy to "vet" what you find interesting---it counts toward my own research, as I believe there is always more to learn about Venice.
Consider reading "City of Falling Angels" by John Berendt if you want some inside stories on this fascinating place.
