Is it possible to install a 64 bit program on a 32 bit OS with a 64 bit processor?
In principle yes, but the processor and the OS have to support it.
On ARMv8, a 32-bit (Aarch32) kernel cannot run 64-bit (Aarch64) processes. This is a limitation of the processor.
There are other processors that don't have this limitation, for example it is possible to run x86_64 processes on top of an x86_32 kernel on an x86_64 processor, but few kernels support it, presumably because it's of limited utility (mostly, you save a bit of RAM in the kernel by making it 32-bit). Linux doesn't support it, but Solaris does.
You can keep your existing 32-bit OS if you run a 64-bit kernel. An Aarch64 Linux kernel can run Aarch32 processes. Raspbian doesn't support this out of the box, so you'd need to maintain both a 32-bit OS and a 64-bit OS. You can use either one as the main OS (i.e. the one that runs init and system services) and the other to run a specific program using chroot. See How do I run 32-bit programs on a 64-bit Debian/Ubuntu? for a practical approach.
Note that you will need to install all the libraries that the 64-bit program requires. Any given process must be either wholly 32-bit or wholly 64-bit, so you can't use a 32-bit library in a 64-bit executable.
Unless you have strong reasons to keep a 32-bit system, if you need to run a 64-bit executable, it would be easier to install a 64-bit system.
Note that the only thing that 64-bit programs can do but 32-bit programs can't is address more than about 3GB of virtual memory, which is of limited utility on a system with 1GB of RAM. You may get performance benefits from the extra, larger registers, but you'll also lose performance from the extra memory accesses.