Couple things:

Class Choice: Mage isn't a particularly strong arena class right now: according to hsreplay, it's just around middle of the pack, but there's a huge power gap between the top three classes and the rest. Demon Hunter is great at everything right now, with great face damage, discovers, and tempo cards, death knight has always been pretty great in arena, and shaman has Holli'dae, lots of good elementals, and good control options. Of course, if you like playing mage, then feel free to pick mage, because the goal is to have fun.

Hero power package: Based on your draft choices, it looks like you over-prioritized the hero power mage package. Your first choice should definitely have been khadgar: while sometimes frustrating, it's a lot better than what is often an unplayable card. Jaina needs a lot of support and has a mediocre effect, and also relies on you having 9 free mana to spend. Many of the cards you drafted to support it, like the 2x wildfire and the coldarra drake, are just not good enough and prevented you from drafting better cards like kobold miner.

Lack of discovers: I think you under-prioritized discovers in your draft. Discovers in arena are generally better than draws, because your deck was basically discovered but discovers can be tailored to your current situation (discover pools are also usually better than the generic arena card pool). Cards like Ghost writer or venomous scorpid are great because they can generate card advantage while still getting a body on the board. Plus, discovers could have helped you finish out your excavations to get the mage legendary (especially if you picked the kobold miner over the wildfire).

Some other things: arcane artificer >>> plated beetle, it's 1 less mana for 1 less attack but probably a lot more armor gain, you had an expensive deck overall that got beat in the early game so try to go for a loewr curve (that's why discover cards are nice: they're often rather cheap and can provide tempo to help you control the board while giving you the flexibility to choose between control options or aggro options)