Hi Nimrod
You asked: not exactly limited time-wise - I'm on a pretty tight budget
You are a perfect candidate for enjoying the benefits of Mongolian public transport . It is true that there are no western style bus terminals, there are no signs in English, and the time tables are somewhat flexible. But - once you adapt to the Mongolian style of public transport you will find it rewarding. The people are friendly, and it is the cheapest way to travel.
The public transport system
Buses go from Ulaanbaatar to all Aimags (province) centers. From the Aimag center you wait with others for a van to fill up going to a Sum (district) center. If you attempt to go to a tourist attraction that is not on the public transport routs, you will find jeeps or vans ready to be hired to take you the last few miles to your final destination. Any private car can act as a taxi. See Public Transportation for maps time tables and prices.
Finding partners
You asked: should it be a problem for me, finding other travelers to share a jeep
Most budget travellers find partners once they arrive in Ulaanbaatar.
The best places are to look at the notice boards at the popular guest houses like Golden-Gobi, Zaya, UB, LG, Khongor, and other popular budget guest houses .
You can also look at the boards of the tourist cafes like Amsterdam, Bernard, Subaatar-Sandwiches, and other popular cafes .
Traveling alone
You asked: how common it is to travel alone in Mongolia
It is common. You will find partners as you go along.
September
You said: I plan to be in UB sometime in September
September is the transition period from summer to winter, it marks the tail end of the peak tourist season.
The advantage is that it will be easy to find a place in guest houses, easy to find a jeep driver, less rain and the Gobi is not as hot. The North might get below freezing at night.
Good planing - a year in advance
Have a nice trip
Dan