The Internet is famously slow in Lebanon. In Beirut, as of 2015, it averaged about 3.2 megabits per second, compared to the global average of 22.3 Mbit/s. At this speed, it isn’t technically even considered broadband.
That being said, the internet is perfectly workable for emails and most websites.
Nearly every cafe in central Beirut offers free, decent internet with the purchase of a coffee (and if you need to work for a few hours, the staff won’t pester you to order more). Urbanista is a chain with outlets in Gemmayzeh, Hamra, and the ABC Achrafiyeh mall that offers reasonable internet and decent food. Want more Beiruti vibes? Head for Ta Marbouta or Mezyan in Hamra, or Em Nazih in Gemmayzeh.
If you need to upload or download large files, the Cigar Lounge at Le Grey hotel in downtown Beirut has the fastest internet in town. Try to look reasonably smart when you show up: this is a classy establishment, with prices to match.
For streaming, Netflix almost always works seamlessly, but other streaming services aren’t as reliable. BBC iPlayer and other country-specific services only work if you have a VPN—a virtual private network. (You have to download a VPN from the internet.)
For internet on your mobile, any phone store sells reasonably priced mobile internet top up cards. Look for the logos of the two major service providers, touch or Alfa. Buy a SIM card for about $20 (don’t forget your passport), then buy a top up card for phone credit and mobile internet; 500 MB costs $19.