Running 64 bit programs on a 32 bit system

There is one way. You can use Qemu or a similar emulator to emulate a complete 64 bit host. However this will come with a significant speed loss.

And just to stress that last part a bit:

• Running a hypervisor to run a 32 bit VM on a 64 bit host will let most programs run at almost full speed. (Long long ago I tested and reached 99% speed of the host using vmware workstation 5 and dnetc, which is a typical user space program.)
• Now if you are going to emulate then expect numbers to be the other way around. 99% speed loss would not be surprising.

So, as an answer to part 1: Technically yes. You might not want to though.

or is there a way of upgrading from x32 bit to x64 bit for free.

If your computer has 64 bit hardware (basically everything from the last decade) then you can install a 64 bit OS. DavidPostill's answer is extremely complete on that. Note that this means a full reinstallation. You can not change the windows OS from 32 bit to 64 bit. You will have to completely replace it.

^{Not associated with qemu. It is just the one I first remembered}

As pointed out by Ismael Miguel: As long as your hardware is 64 bit you can use VirtualBox to run a 64 bit OS. Even if you host OS is 32 bit. The documentation for that can be found here.

Is there a way of running a x64 bit program on a x32 bit system?

You cannot (normally) run 64 bit programs on a 32 bit OS.

See the other answer by Hennes for a (not recommended) workaround.

Is there a way of upgrading from x32 bit to x64 bit for free (Windows 10)?

Is Your PC 64-bit Capable?

First thing’s first, you’ll need to confirm whether the CPU in your computer is 64-bit capable. To do so, open the Settings app from your Start menu, select System, and select About. Look to the right of “System type.”

If you see “32-bit operating system, x64-based processor,” this means you’re using a 32-bit version of Windows 10 but your CPU can run a 64-bit version. If it doesn’t say you have an x64-based processor, you have a 32-bit CPU and can’t upgrade to the 64-bit version.

Does Your CPU Have the Required Features?

Some of the first 64-bit CPUs don’t have the required features to run 64-bit versions of Windows 10, even if they are 64-bit compatible. Specifically, Windows 10 requires PAE, SSE2, and NX. Microsoft has more information about this here. PCs will also need support for CMPXCHG16b, which some of the first consumer 64-bit processors didn’t include.

The 64-bit version of Windows 10 will refuse to install if your CPU doesn’t support these things. If you’d like to check before you begin the install process, download and run CPU Z.

“SSE2” will appear under Instructions if it’s supported. “EM64T” or “AMD64” should indicate that PAE is supported. “VT-x” or “VT-d” always indicates that NX is supported.

On some older computers, you may have to enable NX (sometimes called XD) in the computer’s BIOS. If you receive an error about your CPU not being supported during the install process, visit your computer’s BIOS and look for an option titled something like “No eXecute bit (NX)”, “eXecute Disabled (XD)”, “No Execute Memory Protect”, “Execute Disabled Memory Protection”, “EDB (Execute Disabled Bit)”, “EVP (Enhanced Virus Protection)”, or something similar. Look under the “Security” or “Advanced” tabs somewhere in your BIOS for this setting.

Does Your PC’s Hardware Offer 64-bit Drivers?

Even if your CPU and motherboard support the correct features, you might want to consider whether your computer’s hardware will work properly with a 64-bit version of Windows. 64-bit versions of Windows require 64-bit hardware drivers, and the 32-bit versions you’re using on your current Windows 10 system won’t work.

Modern hardware should certainly offer 64-bit drivers, but very old hardware may no longer be supported and the manufacturer may have never offered 64-bit drivers. To check for this, you can visit the manufacturer’s driver download web pages for your hardware and see if 64-bit drivers are available. You shouldn’t necessarily need to download these from the manufacturer’s website — they should be included with Windows 10 or automatically downloaded from Windows Update. But old hardware — for example, a particularly ancient printer — may not offer 64-bit drivers.

Either way, you can find out for sure by installing the 64-bit versions of Windows. if it doesn’t support your CPU, it will let you know. If hardware drivers aren’t available, you’ll see after installing it and you can downgrade to the 32-bit version of Windows 10 if your hardware doesn’t work.

Do the Clean Install

You’ll need to perform a clean install to get to the 64-bit version of Windows 10 from the 32-bit one. There’s no upgrade path.

Warning: Back up your important files before continuing. This process will wipe your installed programs and files.

First, if you haven’t upgraded to Windows 10 yet, you’ll need to use the upgrade tool to upgrade. You’ll get the 32-bit version of Windows 10 if you were previously using a 32-bit version of Windows 7 or 8.1. But the upgrade process will give your PC a Windows 10 license. Be sure to check that your current 32-bit version of Windows 10 is activated under Settings > Update & security > Activation.

Once you’re already using an activated version of Windows 10, download the Windows 10 media creation tool from Microsoft. If you’re using the 32-bit version of Windows 10 at the moment, you’ll have to download and run the 32-bit tool.

Select “Create installation media for another PC” and use the tool to create a USB drive or burn a disc with Windows 10. As you click through the wizard, you’ll be asked whether you want to create 32-bit or 64-bit installation media. Select the 64-bit architecture.

Next, reboot your computer and boot from the installation media. Install Windows 10, selecting “Custom install” and overwriting your current version of Windows. When you’re asked to insert a product key, skip the process and continue. You’ll have to skip two of these prompts in total. After you reach the desktop, Windows 10 will automatically check in with Microsoft and activate itself. You’ll now be running the 64-bit edition of Windows on your PC.

if you want to go back to the 32-bit version of Windows, you’ll need to download the media creation tool — the 64-bit one, if you’re running the 64-bit version of Windows 10 — and use it to create 32-bit installation media. Boot from that installation media and reinstall the 32-bit version over the 64-bit version.

Source How to Switch From 32-bit Windows 10 to 64-bit Windows 10