If you know skills you are looking to learn for then go ahead.
MBA on other hand is a practical program, where you related and hone your practical skills, unlike other academic programs where theoretical knowledge is basis of learning new skills.
1. It's much better to get the experience first, it will allow you get more exposure of real world business. Once you have that, first, you'll gain more from the program, secondly, you'll be able to contribute to the class.
2. Getting Business Analyst position should be possible without an MBA.
3. Strategy consultant is wider term, depends what kind of strategy you want to get in viz., Market expansion, new product launch, etc. It could be IT strategy as well. To differentiate IT Consulting can be part of Strategy consulting, but not vice versa.
The way it sounds, IT Consulting will be your better bet.
My two cents: Start working, if you don't like it, you'll know and if you do you'll know as well.