I usually aim for 2 extra days pre-tour and it has worked well for me. I think a great deal of that depends on your comfort level for exploring on your own and your budget.
I am a solo traveler as well, and I find that I really enjoy RS tours. I do get the single supplement and am happy that I do. Sometimes it is just nice to have your own space to hang out in for a bit. The people that I have encountered in RS tour groups are generally very friendly and a pleasure to get to know. I find that the idea of "shared experience" with others is a component that adds to my travel enjoyment.
Tips that I have for you include being prepared for ideas on the free time. Have a list of ideas ready of what you might enjoy doing in the free time offered on the tour. You may find some others that are looking to do an activity together or you may find that you want to do something independently. Both are just fine, but having those ideas thought about in advance is really helpful.
Don't hesitate to let people know if you would like to have someone to eat dinner with on a non tour dinner night. The secret here is to find the balance between letting people know you would love to have company and making sure that you respect everyone's right to do their own thing. So far, there have always been people interested in sharing a meal on the times that I have asked. One tour had 7 solo travelers on it and it was very easy to find someone in that situation. Don't expect that you will be invited if you don't communicate.
Finally, don't hesitate to sit down at a table for "4" at a group dinner. My first tour, I didn't feel like I could this and I ended up at the teen table most of the time. Now, fortunately I like teens and they were a fine group. However, I learned that I didn't need to relegate myself away from eating with couples just because I was solo. Ideally, there are tables set up for larger groups than 4.
I think you will really enjoy yourself. My first tour was an experiment to see if I would like traveling in a group. I discovered it was great. I still travel independently at times, but I do enjoy RS tours.