32-bit to 64-bit upgrade

The Vostro 420 is likely out of warranty... Dell are phasing out physical OS media requests and Microsoft have not made Windows 7 OEM Downloadable... Your only choice for Windows 7 is to look in Unofficial Locations for the Dell Windows 7 Professional with SP2 Reinstallation .iso or the likes of eBay which is also Unofficial. 

You can however Download Windows 10 TH2 64 Bit Installation Media from Microsoft. Windows 10 TH2 installation media will accept your Windows 7 Pro OEM Product Key. See here for details:

http://dellwindowsreinstallationguide.com/download-windows-10-oem-and-retail-iso/

https://www.youtube.com/playlist?list=PL1RkaknDn7v-Ucth4gt0U3BHVSY7oNkWr

P.S. Now is a good time to get a SSD before performing Clean Windows 10 Pro 64 Bit installation. It'll significantly speed up that system.

I really don't care that my Vostro 420 is out of warranty, or that Windows 7 is not downloadable, or that I will have to purchase the 64-bit version at eBay.

My question was if the 64-bit version would run on a desktop that came with the 32-bit version.  Can anyone else answer that question, or must I look elsewhere for advice?

P.S.:  I installed a solid-state hard drive more than five years ago on the Vostro, and it did NOT significantly "speed up" that system.  Also, I have Windows 10 on four other computers here in my office, and I don't need another instance of that OS, thank you very much.