On one the popular way for visiting Mongolia is to take a train. You can take the Trans-Siberian Railway and arrive from Russia or China. It’s also possible to take a bus from Russia or China. However, if you choose a train or a bus, then you need to arrange more visas. If you’re anyway visiting these countries, then traveling by train/bus sounds like a good idea.
You can also simply fly to the Mongolian capital – Ulaanbaatar. I was flying to Mongolia from Europe and didn’t have any problems. But be aware, that flights may be delayed or cancelled because of the bad weather, what apparently is not a seldom case.