Update Entity Framework 4 to 4.5

Question

On process/general approach to this:
Well this is slightly unclear question. If you have "production environment" then it implies that you have test/UAT environment, is'n it?
Silly option is just to keep your fingers crossed and do everything on production (see below on how to mitigate this). But it is wiser to sort out your test environment situation. Quite often you can get away with skipping testing of your change, even when introducing more drastic changes, but I guess you want to minimize the risks, right?
If you don't have test environment - you can try and create one, e.g. either by creating VM copy of your production through VM capture process or by recreating it manually as VM. It's perfectly doable thing especially when your app is not multitier one and you just need on test box (not multi-server environment). And in case of budget constraints scaled down (in terms of performance/redundancy) version of production will do for your test.
If you not going to invest your time or other resources into building test environment (even though this is AAA rated high return investment :) ) workout rollback plan for your change, backup everything and doublecheck that you can restore it (not only in theory, but in practice too) - though if you approach to this seriously it's almost equally time consuming as building test environment.
And in properly managed environments you should have all of this: your change should went through test environment and introduced in controlled manner with roll back plan in place.
On updating Entity Framework 4.5: this stuff usually backward compatible & allow multiple version to coexist, but research & test additionally. It all depends :)