I'm only guessing that your Intel i3 uses the same heatsink as the optional i5 and i7's do.
With that being said, you shouldn't have any problem upgrading using the heatsink that came w/your i3 for the i5. Just make sure it is clean of any dust in the fins.
Just make sure to apply some good thermal paste like Arctic Silver 5 for example, and apply it sparingly, since too much can be as much a problem as too little can be, especially w/Arctic Silver, which contains silver, which is a conductor of electricity, so be careful applying it.
Clean both the surface of the heatsink, removing any old thermal paste, using some isopropyl alcohol and a Q-Tip or isopropyl alcohol pads, and the same on the CPU, letting both dry.
On the Arctic Silver's website they have directions on just how to apply it properly.
http://www.arcticsilver.com/pdf/appmeth/int/vl/intel_app_method_vertical_line_v1.1.pdfAs for disassembling in order to get to the CPU, I did a write up w/some pictures, as did some others, just do a search.
I might recommend upgrading your memory at the same time if you haven't already done so, since it's pretty much the same process of disassembly.
The same goes for adding an SSD as well.
Good Luck!
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. ROG Zephyrus GA401QM (Q14) Laptop, AMD Ryzen 9 5900H CPU; Nvidia RTX 3060 GPU; 24 GB DDR4 (1x8 GB, 1x16); 1 TB Samsung Pro980 M.2 SSD; Windows 10 Pro