All hospitals, with the exception of some private clinics, are open to all insured patients. Normally doctors transfer patients to a hospital and control of treatment is then managed by a doctor there.
Hospitals in Austria aren't cheap! All fees have to be paid for either by you or your insurance company. Even in the case of an emergency, there is no such thing as "free treatment". If it is your decision whether or not to go into hospital, check your cover with your insurance company.
When going to a hospital or clinic, take proof of your health insurance with you (although you're unlikely to be refused in case of emergencies). If you are admitted to hospital, your health insurance will cover the costs of that stay.
The room you get depends on your health insurance scheme. If privately insured, you will probably get a single or double room. If state insured, you will probably share with two or three others. Bring a night-dress or pyjamas, towels and toiletries to the hospital.
All hospitals run a special accident and emergency department ( Notaufnahme), where you go if you need urgent and immediate medical attention at any time.
Helpful links: www.gesund.co.at english