Under Adobe's new subscription service where you actually renting an application month to month? yes ,providing your equipment is capable of running of such.
Usually when you buy a suite, once you do, you have to update the suite. And, Adobe actually frowns upon purchasing just what you need. And in their pricing Policies the sum of the parts ends up more expensive than buying the suite. Its like when your tire blows out on your car rather than buying just a replacement tire or set of tires you forced to buy a whole comple Licoln Town Car.
Can I download and run Premiere CS6 when all other programs on my computer are CS4?
To answer your question directly, YES you can install Premier CS6 on top of your CS4 because these packages reside in their own folders so complications are not likely to occur.
As to pricing of the product, you don't qualify for the upgrade price on a single product unless you have a relevant qualifying product for it. This means you must have Premier CS5.5, Premier CS 5 or Premier CS 4 SINGLE products. The upgrade goes like with like and cannot be mixed and matched.
I hope this answers your questions but if you have a follow up question then please post here: