Although the cost of life in Mongolia isn't very elevated, it can be quite expensive to travel around. The budget you'll need will depend entirely on your style of travelling.
The cost of public and private transportation
Public transportation in Mongolia is rather cheap. I didn't travel much by bus, but the ride from Terelj National Park to Ulaanbaatar for example, which is around 65 kilometers, cost less than 2 euros. The reason I didn't use many buses, though, is because there aren't many roads in Mongolia, so if you want to visit remote places, it will be difficult to get there by yourself. Also, the buses don't go very often, so make sure you plan ahead.
If you want to visit remote places, you have a few options:
Travel to Mongolia
If you travel from far, meaning transport by bus or train or not possible, you'll be looking at flying into Ulaanbaatar, the capital of Mongolia. When I wanted to book my ticket, though, the tickets were very expensive, so I compared the prices of the surrounding airports and ended up booking a ticket from Brussels to Beijing, which was much cheaper at 388 EUR for a one-way flight, which I booked from the airline Aeroflot. This allowed me to visit a part of China before continuing my trip to Mongolia by train.
The cost of lodging in Ulaanbaatar
There are several hostels to choose from, most of which will cost you between 7 and 10 usd per night for a night in a dorm. Beware though that for Naadam Festival, the hostels get booked up quickly and prices will slightly rise.
The cost of dining out
It is not expensive to dine out in Mongolia. In the capital, you can eat (a lot) for less than 10 dollars, drinks included. Street food of course will even be even cheaper.
One thing that struck me when buying food in the supermarket, is how expensive fruit is. Everything else is relatively cheap, but for fruit you'll pay twice the European price. It is very likely, though, that you'll reach that point when you're missing healthy food so badly, that you would sell your soul for some nutritious vitamins.