"If she starts in June she'll earn approximately £10,800 in the remaining 10 months of the 2012/13 tax year. The personal allowance is £8,105. She'll pay 20% income tax on what's left - £539 - and £385 in NI. So she should get around £988/month if her tax code is correct."
No, thats not how PAYE works. She is paid £1,083 a month and the tax allowance is £675 a month meaning that normally she will pay £82 in tax and £54 in NI, leaving £948 a month. But in June she will get 3x£675 allowance so won't pay any income tax, in fact her income won't have 'caught up' with the tax allowance until September when she gets 6x£675 allowance = £4,050 and will have earnt 4x£1,083 = £4,333 so will pay £57 of income tax.
National Insurance however does not work on a cumulative basis so your calculation is wrong there too.
So her take home will be £1,030 in June - August, £972 in September and £948 in October-March.
"If her employer puts her on an emergency tax code, her tax bill could be higher to start with."
No, the correct emergency tax code for someone with no other job since 6 April 2012 is 810L which will give her the tax calculations above. It is possible that the employer does not follow the correct procedure (she should make sure that the employer asks her to complete a P46 form before she is first paid) which would lead to 20% tax being deducted on all of her pay, but then it is possible that the employer does not follow the correct procedure and forgets to pay her at all - anything is possible, but not necessarily helpful to know!
I could have done without taking the time to post all this detail, I don't know why people think it is helpful to post things when they don't know what they are talking about, particularly when a correct answer has already been given by someone who does know what they are taking about.