You can call +39 055 294883 to make a reservation in advance and walk right in. The phone operator will give you an extension number which you quote at Gate 3 to pay (cash or credit only) and get the tickets. Alternatively, there is a ticket office at the museum which will normally sell these same reservation tickets for almost immediate entry. Online booking is available but is much less convenient because it has a 24 hour waiting period, your specified time may change and you need to print an email.
The restaurant/caffè has a large balcony overlooking the main piazza with good views of the Palazzo Vecchio. It is a great place to take a break for art lovers making a non-rushed visit to this fantastic collection. This cafe is rather expensive however. Street performers are often seen outside the Uffizi.