Believe it or not, Saransk does indeed have an international airport. The imaginatively named Saransk Airport went under extensive reconstruction in 2017, ahead of an expanded schedule in the World Cup year. The airport is just 7km from the city centre, but the only way into town (at time of writing) was via taxi, so best get one booked ahead of time.
This is Russia, so train is going to be the best bet when it comes to travel. Saransk-I station is a beaut as well, and the city is relatively well connected with surrounding towns and cities. Whether or not it is prepared for the millions of potential visitors during the World Cup remains to be seen.
Saransk has a bus station and a number of connections with nearby towns and cities, but little else to write home about on this front. As with almost everywhere in Russia, best to stick to trains.