Getting connected to the cell network in Belarus is only allowed after presenting identification (a passport).
One option for non-residents is to acquire a local SIM-card by MTS carrier called MTS SMART Guest for 12.5 roubles (~7$).
You can sign up in the airport by showing your international passport or ID. The customer center is opened from 9AM to 9PM.
Detailed location of the center can be seen on this map (red MTS logo near gates 5 and 6)
There is another option from Privet (as suggested by vcsjonesq). Their rules say that non-residents are allowed. They offer four tariff options from which you can choose based on your needs.
Credit:stackoverflow.com‘