Is it worth reinstalling 64-bit Ubuntu instead of 32-bit on old hardware?

It won't be faster in any meaningful way, but the reality of 32 bit is that it's increasingly legacy, which means, in the real world, bugs are not getting found and fixed.

I would install 64 bit if I were you primarily because it will be more reliable and have bugs fixed more often. It will also future-proof your machine, since you can then just update to the next OS version every now and then without having to worry about it.

All FOSS projects have finite developer eyes/hours available to them, and 32 bit just isn't getting many of those eyes and hours anymore, and there are no magic coders that just fix stuff because it's broken, sadly. It's wisest to just accept this limitation as reality, since it won't really change.

I know of at least one 3rd party Linux kernel builder who stopped supporting 32 bit relatively recently because of a series of 32 bit kernel bugs that were known but were not getting fixed, nor did they appear likely to ever get fixed in the future. And that's the Linux kernel project, which has thousands of contributors. It just goes downhill from there with other projects with far fewer developers. This situation will happen to more and more core and not so core software and tools as 32 bit gets removed from more and more primary GNU/Linux distribution pools.

This becomes increasingly relevant as major projects like Google Chrome, Firefox, etc, start to drop, if they have not already dropped, 32 bit support, which means you'll be using insecure non-updateable software to access the internet.

Note that you can in theory sort of cross grade 32 bit to 64 bit (at least on Debian, not sure about Ubuntu), I tested that on one machine to see, but it's such a pain, and takes so long, and leaves so much cruft, and requires so many manual fixes, that in the end, I decided that was not worth it, and just switched the rest of my systems to 64 bit by reinstalling.

Keep in mind you can copy your main configs, and then get a package list, and reinstall the packages when you reinstall to 64 bit, it doesn't take that long, and once it's done, no more need to worry about it.

Your other option is to just never upgrade your box again, and just let it run until it dies. On systems that don't interact with the internet that's not a terrible way to deal with the stuff, but you may hit a snag one day when you need to match versions of something like samba or nfs and you can't because your server box OS is too old.

It's probably worth reinstalling to a 64-bit operating system. As Lizardx mentioned, there's less focus on 32-bit x86 systems right now and people are fixing fewer bugs there.

Additionally, it is a real possibility that it will be slightly faster, since x86-64 has 16 general-purpose registers and x86-32 has only 8. Position-independent code, which is used for security reasons in most binaries these days, is free on x86-64 with RIP-relative addressing but needs an additional register on x86-32. Having additional registers means that programs can keep more data on the CPU instead of having to go to memory, improving performance.

However, there will probably be a slight increase in memory usage, since pointers will be 64 bits instead of 32 bits. If you're really stressed for memory right now, that won't make it any better, but otherwise, it's probably fine.

In general, my recommendation is to do so. You'll get a better supported, slightly more performant environment.