Internet cafes are everywhere in Addis Ababa and other major towns and are fairly easy to come by in smaller places that see few tourists. Most are open 8am to 8pm Monday to Saturday; many open with limited hours on Sunday. Most, however, come and go with monotonous regularity and few last the distance.
In-room wi-fi is increasingly common in most hotels, including most budget hotels. Before you get too excited, remember that internet connections in Ethiopia can be among the worst on the continent as bandwidth is often insufficient, although things are slowly improving.
The Ethiopian government is highly suspicious of the internet. Opposition websites and others critical of the government are frequently blocked. During the unrest in 2016, all social media (Twitter, Facebook, Skype, etc) was completely blocked, and although most Ethiopians managed to find ways around the ban, that option was not easy for casual visitors.