With serially-addressed display drivers you can use as many digits as you want. (Start with one digit and work your way up.)
A count-down timer is simple, and you might by using the serial monitor (so you can start experimenting/programming without a display or any additional hardware).
panel should have a voltage rating, plus a current/and/or power rating. If they don't give you all 3 numbers, Power (Watts) = Volts x Amps. Of course the solar panels specs are likely optimistic.... at
something similar to the solar powered calculators used in elementary schools?
You'll need a bigger solar panel and it will depend on the current requirements for your display. The solar noon with clear skies, in the summer, on the equator. 
You can estimate the current requirements for your project, then measure the current once it's built.
I would like for the battery to last 3-6 months if possible.
Check the milliamp-hour rating on the battery compared to current requirements of your project.
With solar recharging the battery only has however long you expect to go without sun.
You'll need quite a bit more (short-term) power from the solar panel than the Arduino/display requires because you could be charging the battery for 1 day or less and then discharg for a few days. ...It's a statistical game.
If you need accurate time over months, you'll need a real-time-clock module (RTC) because the oscillator built-into the Arduino is not accurate. The RTC won't be perfect either... No clock is perfect, so check the specs to see if it's accurate enough for you. If not, you'll need a cell-phone, Internet, or GPS connection to get the "true-time".[/quote]