Greetings!

Although it took me a couple of tries, I've finally managed to beat Noth the Plaguebringer with a modified version of NiymN's Ramp Druid Deck - I higly recommend that you go and check his version too! As for the proof, here's a screenshot - please don't mind the Polish game client and the fact that it doesn't seem like a decisive victory (I had some bad draws):

List of changes and possible substitutions

• Healing Touch x 1 + Swipe x 1 -> replaced with Starfire x 2 - this is mostly due to the fact that Starfire adds some extra card draw that you might need later on. Moreover, it's also better in dealing with most of Noth's minions, since a lot of them have 5 health and die to this card without having to take damage to the face, as you often had to when using Swipe + Hero Power.

• Ancient of War x 2 -> replaced with Ironbark Protector x 2 - mostly due to the fact that I don't have Ancients of War in my card pool, but I highly recommend that you revert this change, if you do have the Ancients of War in your pool.

• Maexxna x 1 -> replaced with Sylvanas Windrunner - due to the fact that most Noth's minion have 5 health, they can also be killed with Sylvanas. While Maexxna usually trades for 2 cards, Sylvanas allows you to trade for 1, remove 1 by stealing it and trade for 1 with the stolen card. All of that at a slightly lower mana cost. However, if you do not have Sylvanas, feel free to put Maexnna back.

• Frost Elemental x 2 -> replaced with War Golem x 2 - mostly due to the fact that I didn't have Frost Elementals in my card pool. Personally, I think that both options are good and you'd be fine picking either one.

• Sea Giant x 1 -> replaced with Stormwind Champion x 1 - for the same reason as above, however... in case of regular minions, it doesn't really matter which one you pick, as long as they have a lot of health and are able to kill with 5 health minions. Cards like Sea Giant or Molten Giant might be a bit cheaper in terms of mana, but since you won't be hoarding a lot of cards anyway, it's not really that important. That said, feel free to pick any big cards you please.

• Legendaries - while Ragnaros the Firelord and Ysera aren't really fit for a budget decks, they are really, really strong in this encounter and you should avoid replacing them. The rest of the cards is fairly low budget, so provided you have Ysera and Ragnaros, you shouldn't have any problems building the deck.

Strategy and tips

• Mulligan - it is essential that you get a decent starting hand and are able to play Wild Growth as soon as possible (preferably at 2 mana). Cards worth keeping in the early game are: Wild Growth, Claw, Swipe (both for early game removal), Druid of the Claw (cheapest taunt you have) and Innervate. The more of them you get in your first 1-2 turns, the higher the chance you'll end up winning. This is particularly true for Wild Growth and Innervate that allow you to get your big creatures on the board much faster and thus set up more efficient trading scenarios. You might have to conceade a couple of times before you get a decent starting hands + first two draws. 

• Board clear - Poison Seeds + Starfall (for 2 damage) combo is your biggest friend in this match-up, because it allows you to fully clear the board when you're getting overrun. Just remember to sacrificially trade your minions before triggering the Poison Seeds, otherwise Noth will get a free skeleton! A good idea is not to flood the board to much, because you never know when it will be a right time to use the combo. 

• Face/trade ratio - you have to balance it out in a way that allows you to keep doing damage to the enemy, while also setting up preferential trades. For example - if one of your biggest minions is hidden behind taunts and it's likely, he won't be killed next turn, feel free to go for face, but if there's a chance that some low-cost minion will kill anything you place on the board, ignore going for face and trade on your terms (with something bigger). Don't go over the top with the trading though, as you may end up in a situation where you don't have enough firepower to finish Noth off.

• Secrets - two most notable that Noth plays are Duplicate and Counterspell. Deal with the first by killing less valuable minions first and with the latter by casting Claw or Innervate (should you have it) before any other spells, especially when you're planning to use Starfall + [deck]Poison Seeds[/deck] combo.

Good luck!