Uno can do that.
Need 11 current limit resistors.
LEDs & resistors here
http://www.dipmicro.com/store/index.php?act=viewCat&catId=511
Pick resistor based on (5V - Vforward of LED)/.02 = resistance needed.
Browse Learning Section, or the Playground, for ShiftPWM, will let you fade up/down LED on any pin, and not just the 6 pins with hardware PWM.
9V battery only has 200-300mAH of power. Use 5AAs (7.5) into the barrel jack instead, for 2000-2500mAH of power. Add a 6th if you want, will just into heating up the 5V regulator tho.
11 LEDs drawing 20mA = ~220maH by themselves when all are on.