Not really, Rats will take care of the Ruins and your Apprentices will be very happy to see a bunch of Rats.
Probably opening Armory/Market Square, so the Apprentice/MS player has only few MS to get his Golds. If this works it should be easy to win with the vineyard route, since 10 ruins, about 10 rats and 5 MS are already 8 point Vineyards, and Potion can be gained and topdecked by Armory for fast points.
For Apprentice-Market Square you don't really need more than 3 MS, so denying them beyond the point where they are also the best choice for yourself seems pointless.
Only 3 Market Squares? I can buy as many Market Squares as i can and mega-turn out with Apprentice-Market Square if i'm left uncontested.
In a non-mirror, Apprentice-Market Square has very little use for more than 3 Market Squares. The idea behind the strategy is to get lots of Apprentices, draw your deck, then trash a Gold, discard 3 Market Squares and redraw from the trashed Gold your 3 Market Squares + the 3 new Golds. Rinse and repeat as often as you can.
Since you won't be able to gain enough Golds in one turn to pile out the Provinces with a single mega-turn, you'll need at least 2 turns if you need to get all 8. 2 x 4 buys, i.e. 3 Market Squares, is all you need for that. This doesn't mean that you shouldn't be buying additional Market Squares with spare 3s and 4s (though here I'd probably get Wandering Minstrels for $4 past the early game), but it does mean that denying them is pointless for your opponent, as you'll be able to get 3 regardless of what he does, and 3 is all you really need.
Apprentice, on the other hand, is a card your opponent would love to deny, but the only way to deny lots of Apprentices on this board is to go for Apprentice-Market Square yourself.