The best resource on the topic is now the prepaid data sim card wiki.
Mobile internet is cheap. For 10$ you can get a sim card, call and go online for a month. Your foreign sim card will most likely work on roaming in Uzbekistan.
The main issue before was getting a sim card. Telephone companies would not register foreigners, so you needed a friend in Uzbekistan who wanted to register your sim card in his name. Since 2017, this issue has sort-of been resolved. If one company tells you they cannot sell you a sim card, try another company, or a different office of the same company. Should work.
Or not? The rules for buying a sim card in Uzbekistan change frequently and are not applied evenly. Expect everything.
You questions and comments are welcome in the Uzbek sim card forum thread.
With the political thaw under new president Mirziyoyev internet access has improved. Still, a lot of websites can only be accessed through VPN/proxy: news sites, blogging sites, torrent sites, opposition sites, porn sites.
The main social networks remain accessible though.