I have a couple of tattoos, but I'd like a partial sleeve. Maybe a half-sleeve or quarter-sleeve on my upper arm. 
How long does it take to get a half-sleeve done?
Well, there's a difference between an arm covered in tattoos and a sleeve tattoo. A sleeve has a unified theme and may take many sessions to complete. 
How painful is it? I have never gotten one.
Tattoo
A tattoo is a form of body modification where a design is made by inserting ink, dyes and pigments, either indelible or temporary, into the dermis layer of the skin to change the pigment. The art of making tattoos is tattooing. 
Tattoos fall into three broad categories: purely decorative (with no specific meaning); symbolic (with a specific meaning pertinent to the wearer); pictorial (a depiction of a specific person or item). In addition, tattoos can be used for identification such as
Sleeve tattoo
A sleeve tattoo (or tattoo sleeve) is a large tattoo, or a collection of smaller tattoos, that has a unified theme, that covers most or all of a person's arm, usually from shoulder to wrist.
There is a difference between an arm covered in tattoos and a sleeve tattoo. A sleeve tattoo has a unified theme, whereas an arm covered in tattoos may have many tattoos of different styles that does not have an overall unity.
The term "sleeve" is a reference