The first on my list, and my favorite if I want to disconnect from the internet for a while, is Poniu Laime (which I happen to call the 'green door bakery') on the corner of Stikliu g. and Zydu g. They have a huge selection of cakes and pre-prepared Lithuanian dishes that make a quick lunch and coffee (that in all honesty is my least favorite of the cafes listed here - too bitter for me). The best part of the cafe is the huge amount of outdoor seats with umbrellas. Outdoors at this cafe you can feel comfortable sitting undisturbed for hours - perfect. Pro tip: arrive early (before 12) and you will have the outdoor seats completely to yourself.
Just like any other city, Vilnius has a few local drunks and homeless people that can occasionally become pushy, particularly when you are sitting outside. I've never had this problem at any of these three cafes.
How do you take advantage of nice weather when you have deadlines to meet?