This is both not possible, and possible at the same time. Let me explain.
No, you cannot retrieve the updates from your installation of Windows. Well, depends on your version of Windows.
There are 2 options that I am aware of:
Option #1, if you are using an OS prior to Windows 8, then you can download a utility that will allow you to use your original install disc, and it will extract what is needed from your current running OS to allow you to include all updates you've done so far, as well as Service Packs, etc., and create your own streamlined, auto-install disc that will also include your product key (automatically entered for you), and all the drivers for the hardware you currently have installed, if you choose to set it up that way. There are separate utilities for XP and Vista/7. There may be one for 8, but I haven't seen it yet. I'll have to check. 8 re-installs itself pretty nicely without too much issue, though. The one for Vista/7 is called rt-7-lite, I believe.
Option #2 is to get the updates installed by using WSUS Offline Update, which allows you to have all your needed updates burned to a DVD or USB flash drive, and they can be installed from there. The tricky part of this is, you have to download the updates.
However, you don't have to go in an pick and choose updates to do this. You can select your operating system, whether or not you want MS Office included in the updates, etc., and download them all for placing on offline media for streamlining the updates later.
The WSUS Offline Update software will automatically scan your Windows installation and determine what updates are needed, just like the built-in Windows Update feature, and then it will install those updates.
Do a web search for WSUS Offline Update at the site afterdawn.com and you should find what you're looking for.
The downside is that you have to have a good internet connection to get these updates downloaded the first time. You can do like I do and take your laptop, borrow a laptop, whatever, and go to a place that has good WiFi, or go to a friends house and use their broadband and their computer, something.
The Offline Update is generally updated once a month or so, as far as I can recall. I just used it this week to re-install XP on someone's laptop, and Vista on someone else's machine. Nice tool to have on hand when you help people fix their machines.