You should not get taxed for the whole amount. You will only be taxed on anything over £12,500, like Kpowell has said this is usually worked out by the firms accountants when you first start, it's possible they'll stick you on a temporary tax code until it is all formalised.
To work for £13,000 would still be worth it, the tax you pay on that extra £500 is so small it'll still be beneficial I would of thought, unless they are demanding a lot more responsibility/hours from you.
National Insurance is still charged on everything you earn, no escaping that one sadly.