Van Allen belts were discovered/confirmed by the satellite "Explorer", which was launched after Sputnik. If so, was Russia aware of VA radiation belts when they launched Sputnik? If not, how did they overcome the radiation problems, if any?
Sputnik-1 had initial orbit with 950 km apogee an 220 km perigee. It was planned to have 1450 km apogee but the rocket underperformed slightly.
950 km is far from most intensive radiation belts.
Also, Sputnik-1 only operated for 22 days before its batteries ran out. So it probably hadn't enough time to suffer significant radiation damage.
Remark: initially Soviet specialists planned to launch a heavier satellite with scientific equipment but it fell behind schedule. So the scientific launch was postponed and Sputnik-1 was designed and launched to guarantee it would beat an American satellite launch.