The Grange City Hotel's location is excellent. We stayed at a hotel a couple of blocks from it on our May 2016 trip to London. We admired the hotel (very nice), and walked behind it to get a look at the Roman Wall that runs behind it and beside of it.
If you walk two blocks south of your hotel, you are at the Tower of London and Tower Bridge. Near your hotel you have several great transportation options:
1. the tube; Tower Hill tube station is a block and a half from the front door of your hotel.
2. the number 15 bus, which runs along Lower Thames Street all the way to Trafalgar Square and the theatre district. The number 15 bus, when traveling west, will take you to St. Paul's Cathedral. Across the street from St. Paul's is the Visitor (or Tourist) Information center for London. Very useful. Walking north from St. Paul's a short distance is the Museum Of London, a terrific must-see.
3. Thames Clipper riverboats dock at a pier right at the Tower Of London. This is a great way to travel in London! Just get on the boat, travel west, and you get off on the dock right at Big Ben, Houses of Parliament, and Westminster Abbey.
Travel east on the boat, and you will soon be at Greenwich.
The Grange is an excellent hotel in an excellent location. You said you will be going a few days early before the tour starts. If that is two days, then the extra money the hotel costs will not be that great. However, if you were going to be there several days before the tour begins, say 5 to 7 days, then the dollars would mount up.
My personal preference would be to stay at the same hotel the tour would be using.
You also said this is your first trip to London. One thing that has helped me on my trips to London, is to stay at the same hotel the whole time I am in London. That way, you get your bearings around the city from this location. You become familiar with the neighborhood, the streets near your hotel, and the way the transportation systems near your hotel work. It is sometimes disorienting to me to change hotels halfway through the trip.
Here is the map page from your hotel's website:
https://www.grangehotels.com/hotels-london/grange-city/local-attractions/
As you can see, you will be very near the Tower Of London and the Thames River.
I would try to find a "deal" on the hotel's website for the days you wish to reserve. It may be worth calling the manager of the hotel and explaining that you will be with that tour group, and wish to stay a couple of days beforehand. Chances are, he or she will give you a good deal. Plus, October is off-season. Most tourists will be gone. Prices for hotels in London go up during the tourist season (summer) and down in the off season (fall and winter). The exception is special events, which always drive the price up.
If you still want to explore another cheaper hotel, Chris's suggestion of looking at Premier Inns is a good one. They are known for good quality, clean rooms at a bargain price.