Do I need more RAM or a graphics card?

You shouldn't need to upgrade the video card. Only time you need to do that is if your running a game, video, or something that requires HD pic or video. More RAM is what you need.

Excel doesn't make use of graphics cards for actual data calculation, just cosmetics.

8GB is plenty of RAM. You can double check Task Manager to see whether Excel even uses that much RAM when working with the file.

I think you should do something about the file instead, maybe break it into more manageable pieces, separate sheets that don't need to go together, combine data from sheets that don't need to be split apart, or fix any unwieldy formulas that take too long to process.

The question to ask first is why do you need to copy/paste such a large data set? I can't conceive of anything that would require you to copy and paste over a Gigabyte worth of Excel data…

Upgrading the GPU or RAM isn't going to fix your problem as its not an out of memory issue that you have as the computer would just have swap the data to the hard drive for temporary storage. That's why you need to go through the system shutdown process and can't just randomly turn your computer off.

Since you did not state what data you are copying and pasting, I would assume that you are able to output that data into some kind of container, perhaps a csv, file? And then from there, merge it into the Excel spreadsheet?

Either way, you need to figure out how to merge the new data into the existing Excel datasheet instead of copying and pasting the values. That would solve your issue without having to resort to upgrading your computer. Beside, with only two DIMM slot to work with, it can get pricey real quick. And a discreet GPU would not do anything for you.

RAM