An Arduino with a some extra bits and pieces would do nicely. Generally if you are making digits on LEDs you will use a 7 segment display. A quick peek at common large seven segment LEDs (futruelec.com) shows that they are made by putting a bunch of LEDs in series. This is good and bad. You won't need a lot of current, but you will need 9.25 volts for a 4 inch tall display. You only get 5v out of an arduino pin.
The easiest solution might be:
1) Stick with a 2.3" display. This only needs 7.4v@20ma.
2) Use the raw power pin of the arduino, this is called 9v but is probably more like 10 to power the LEDs.
3) Use a (9-7.4)/(0.020) = 130... a 150 ohm resistor in series with each segment to keep the current under 20ma.
4) Use the LED driver tutorial... http://www.arduino.cc/en/Tutorial/LEDDriver
5) Power the 4794 chip from the 5v power, not the raw!
6) If you use the decimal point it will need a much larger resistor, it has half the voltage drop.
From there it is just a matter of blinking the lights.
You could also just use some cheap tiny transistors as switches instead of the 4794, but then you'd need to use all 13 of the digital outputs (stay away from digital0, you'd need to move a jumper after programming, and you don't need the upper lefthand segment of the lefthand digit for numbers 1-30) and you'd do a lot more soldering. And you know someone will want to put a 4 or a zero in that lefthand digit as soon as they see it.
If you get seduced by the giant 4" leds... you might give it a go with the transistor method and leave out the current limiting resistors. It looks like the so called 9v DC power supply probably puts out enough voltage to light them without putting out enough to fry them. But you never know about wall wart power supplies. Some of them put out quite a bit more voltage than they say. You could use a regulated 12v supply with a 150 ohm resistor too.