Can I run 64 bit ubuntu on my pc (>10 years old)

Intel’s summary of your CPU’s features confirms that it supports 64-bit mode, as indicated by

CPU op-mode(s):      32-bit, 64-bit

in lscpu’s output.

This isn’t an Atom CPU either, so the rest of your system is, in all likelihood, capable of supporting a 64-bit operating system.

You can re-install a 64-bit variant of your operating system, or you could use Ubuntu’s multiarch support: install a 64-bit kernel, add the amd64 architecture, and you will then be able to install and run 64-bit software without re-installing everything:

sudo dpkg --add-architecture amd64
sudo apt-get update
sudo apt-get install linux-image-generic:amd64

(followed by a reboot).

Here is a guy with the same CPU running 64-bit Ubuntu:

So the answer is yes: You can run 64-bit Ubuntu on your CPU.

Yes, it will work.

E5300 is an x86-64 CPU, and AFAIK no vendor has made a motherboard or laptop for an x86-64 CPU that somehow stopped an OS from switching into 64-bit mode after the kernel loads.

You might have to boot with pure legacy-BIOS 16-bit booting, not UEFI, but once the kernel loads it's only limited by the kernel's drivers and what the CPU can do.

Should I buy a new pc?

Yes, if you want good desktop GUI performance, or any kind of number crunching. Also yes if you care about energy efficiency; modern systems idle at significantly fewer watts than an old Core 2.

E5300 is 2nd-gen (45nm) Core 2 (Penryn / Wolfdale) which should help the power budget some compared to an old E6600 I have lying around as a home server. (And 2nd-gen means that it has SSE4.1 SIMD extensions, and its SSSE3 shuffles are faster, which matters for a few tasks, e.g. video encoding.)

Memory bandwidth is much better in a new system (AMD Zen 1 / Zen 2 or Intel Skylake-derived) compared to Core 2 Duo (like factor of 4). The clock-for-clock single core performance is also probably close to a factor of 2 faster in a lot of things even when not memory bottlenecked, probably including running a web browser.

And there's also the ratio of clock speeds between that 2.6GHz Core 2 vs. a new CPU that can turbo up to 3.5 or 4.5 GHz (depending on CPU model) for bursts of performance (like rendering a web page), so you're looking at a large performance gap, definitely noticeable for interactive use.

Even with 8GiB of RAM, that Core 2 system won't be nearly as snappy for web browsing as a new-ish system based on at least Haswell, preferably Skylake or Zen.

I used a Core 2 Duo E6600 (first-gen 65nm Core 2) with 5GB of RAM at DDR2-533 for a while until about Dec 2016, and it was really not fun to have a lot of tabs open in Chromium. Even without swapping, it was pretty laggy.

Think twice before investing money into new RAM for that old clunker.