Firstly, no, you don't get anything when you apply for a work visa, except the work visa.
Secondly, yes, you will get healthcare automatically when you arrive, but that has nothing to do with a National Insurance number.
Thirdly, there are two separate numbers being talked about here, which are often confused with each other:
- An NHS number is assigned to you when you register with a an NHS doctor in the UK. It is a number that is used to track your medical history when you are treated in the UK. Everyone is assigned one when they are entered into the NHS system, whether they are entitled to NHS care or not. So, once you arrive in the UK, you will need to go to a UK doctor's surgery and register and you will be assigned an NHS number.
- A National Insurance number is used to work out the correct amount of tax you need to pay on your income. It is completely separate from an NHS number. You cannot apply for one until you are in the UK and you may need to apply in person at a Job Centre. You can start work before you have it, but you will be taxed at the emergency tax rate until you have it. See here for how to apply for one:
http://www.hmrc.gov.uk/ni/intro/number.htm