Alright, it appears that I'm 14 pages late to this party. But that's okay I hope some people value my opinion and I apologize if I repeat anything that's been said about this card already.
Initial Thoughts - Wow, $8 is a hefty price tag. A one-shot that eats an action from your deck allowing you to play said card at the start of every turn. I can see why it gets this expensive.
The penalty for having Prince miss an action is devastating. Depending on how large your deck is it could be some 3-5 turns before you get to it again. In which those turns could all have been vastly improved.
The $4-cost limitation puts a clamp down on all of the high powered $5-cost cards. Obviously Bridge, Princess and Highway circumvent this but in order to do that you need cost reducer, Prince and said $5-cost card. While extremely powerful, this will be difficult to pull off and both players can go for it so I see no issues with balance.
In regards to cards played, a cantrip is essentially +2 Actions, +2 Cards (and whatever other bonus it had). Villages become akin to +3 Actions, +2 Cards. Draw is similar to +1 Action, +X Cards. Smithy would be like playing 3 Labs at the start of your turn. Attacks seem slightly less desirable as they have much less benefit to your own turn. Cursing attacks would most likely already have depleted Curses so they are weak. Militia by far seems to be the strongest $4 or less attack to hit.
This is clearly both an engine card and slog card. In an engine, the high opportunity cost warrants large payload. Especially because Prince does nothing to give it's own payload. That needs to come from other cards. What you want Prince to do for your engine is to give you the power to get going. Starting each hand with either a Village or Smithy will do miles for your engine reliability.
The difference between this card and the other high cost engine powerhouse (KC) is that KC promotes explosiveness whereas Prince gives powerful reliability. KC can overcome huge deficits by making huge trees of actions. Prince does not have that ability. Instead of giving you a few humongous turns Prince will give consistently better hands. Prince needs to insure that engines start out with a good hand that is capable of better hands.
While KC will attempt to end the game all at once a Prince player wants to extend the game. If you have an advantage on Prince'd actions then each extra turn of the game will give the better Prince player an advantage. So while KC usually ends in a few explosive turns, Prince will do better with extending the game via alt-VP. Colonies help this as do other things like Fairgrounds, SR, Gardens, etc... It's in this paradox where I believe Prince will shine best. You initially build up a powerful deck but then extend the game to extend Prince's benefit to your deck.