I don't know enough about The Shadow to know the answer to this question, but are the ring lamps controlled by the lamp matrix?
If not, none of the following info will be relevant.
It's not AC at all. Both the stock game and LED OCD output a pulsed DC to the lamp matrix. Both are at about 18V DC. The difference between the two is that LED OCD pulses them at a faster rate and varies the pulse width to control brightness. Full brightness, for both the stock case and with LED OCD, is 18V DC that is on about 1/8th of the time.
Rectifiers are not useful other than making it such that the polarity of the LED doesn't matter. If you turn the LED the right way, you won't need a rectifier.
The resistor value will depend on the specs of your LED. Since they are being pulsed, you can usually get away with driving them with more current than the rated continuous current in the LED specs. Some will have a maximum pulse time, others, you'll be guessing.
If you assume that you don't want to ever overdrive them, you will need to work with 18V as your supply. Pinball LEDs do not do this. They assume ~6V, which is what the incandescent bulbs are rated at.
If you want to make sure the LEDs do not fail prematurely, look up the typical operating current (often 20mA) for the LED and the forward voltage (1.2V is a good guess) at that current. Then use Ohm's law.
V = I * R
18V supply minus the drop over the LED = the current you want to run them at * the resistor value.
In my example
18 - 1.2 = 0.020 * R
R = 840 ohms
If that's not bright enough and you don't mind risking the LEDs getting burned up, you could calculate for a 6V supply, taking the pulsing into account.
6 - 1.2 = 0.020 * R
R = 240 ohms