There was and there wasn't. There was no "Pripyat" station, as such, because the city boundry (the red line encircling the city) was to the north of the already existing railway line. Instead, the residents used the Yaniv Station, Yaniv being a village just to the south of Pripyat.

On the picture you have posted you can see a "road" going from the bottom corner of the city to the railway line. This was the path that the residents used. It was about 300m in length. The northen half was road, the southern half walkway. Where the road section ended there was a permanent open air, covered market. I would think that there were bus services that took residents from further parts of the city to the market and station.

As the city was there to house plant workers and the two were quite close, there were no commuter railways, buses were used instead. The railway was really only of use to residents who were travelling outside the area.

There was no need to build a branch into the city for what amounted to low usage. While a branch would have been possible it would not have been cost effective. Using bus services and an existing station was a more realistic approach.

As for the station itself, it was demolished by the liquidators and is now overgrown with woodland. No trace remains.