Yes, we have this great thing called public transport that makes it very easy to get around without a car. The central parts of Norrköping are all within walking distance from the railway station, but if the weather is horrible or you want to get further out there are both buses and trams.
The things you would want to see are within short walking distance (max 10–15 mins) from the railway station. Bring an umbrella and you’ll be fine regardless of weather.
The first time I went to Norrköping I did exactly the same and I didn't even use public transport, so it's perfectly doable and very pleasant too. Only the rock carvings, should you want to see them, are located a bit further away, but still walkable distance.