I have an iMac7,1 Intel Core 2 Dup 2 GHz 1GB memory 4MB L2 Cache 10.6.8 Can I add memory to update software

There are two iMaci7,1: a 20" mid 2007 and a 24" mid 2007. Both can upgrade the memory to 6 GB total. Yours appears to be the 20" mid 2007 model.

It can utilize 6 GB of memory. That would be a 4 GB module and a 2 GB module. This Apple document describes how to install memory in that model: iMac (20-inch, Mid 2007)

The only two suppliers I would recommend for RAM is Crucial.com and OWC (MacSales.com). This information is from the free app MacTracker.

Your model can run macOS 10.11.6, El Capitan.

If this is yours:

https://everymac.com/systems/apple/imac/specs/imac-core-2-duo-2.4-20-inch-aluminum-specs.html

you do indeed have only 1 GB of RAM which is enough to run Snow Leopard, but nothing further. You can add some RAM, but you need to remember that this machine is from 2007 which makes it about 14 years old. That is ancient for computers and technology. That also means that it is sorely outdated and, in my opinion, not worth updating or putting money into it.

You can upgrade the OS, but it will not run satisfactorily on the RAM it can handle.

Whether you can add memory (RAM) depends on the year and model of your Mac. Some macs can have memory user changed or added and can not as it is hard wired. Please tell us what year and model Mac you have. Changing your operating system is unrelated to your memory. You say that you have "1GB memory". No, that is 1GB of storage. As I said memory is RAM. If you wish to increase your storage, that will involve changing out your hard drive, which the average user should not attempt to do. Now, for the last issue. Your computer says that its OS is up to date because it is up to date. An update is when the same operating system is updated with a newer version. An upgrade is when an operating system is replaced with a newer OS, such as upgrading from Catalina to Big Sur. They are two different things.

An iMac 7,1 is a Mid 2007 model and can officially handle up to 4GB RAM. Unofficially, it can do 6GB but the special RAM module that allows that party trick is quite expensive, costing up to 3x per GB more than regular RAM. For the OS versions your Mac supports, max of 10.11 El Capitan, 4GB will work fine.

With 4GB RAM aboard you can find the El Capitan installer link in this Apple article:

How to get old versions of macOS - Apple Support

It can install directly over Snow Leopard 10.6 without intermediate steps.

Even with OS 10.11 you will not have the latest Safari and you will encounter web pages that will not work. FireFox still supports macOS 10.1—for now— so it is good option.

I buy Mac RAM ONLY from Crucial.com or Other World Computing. This is the latter's RAM listing for your iMac:

https://eshop.macsales.com/shop/apple/memory/iMac/Intel_Core_2_Duo

Note the huge difference in price between 4GB and the 6GB option:

I would invest $20 in a 2007 Mac I wanted to keep in service, but not nearly $80.