There are plenty of Wi-Fi spots around the city where WiFi is free of charge. Instabridge is a Stockholm-based company that can help you find free WiFi, the app is available for iOS and Android.
Here is a selection of locations with free WiFi:
Stockholm Visitor Center
Name: “Stockholm Public”.
Stockholm Arlanda Airport
Name: “Airport-Guest” (Free for three hours)
Central Station Stockholm and Cityterminalen
Name: “All Station Guests”
Free WiFi for guests is very common in cafes and hotels in Stockholm. For example coffee chains like Waynes Coffee, Espresso House and Pressbyrån or hotels like the Radisson Blu, Scandic Hotels, First Hotels and Nordic Choice Hotels, to mention a few. Ask the staff for details.