Hard Disk Monitoring

I would like to monitor the hard drives in my NAS and am following the guide below -
https://www.hdsentinel.com/nas/monitor_hard_disk_status_of_synology_ds220_plus.pdf

I am unsure which Linux installer to use for my Synology DS1821+ NAS device. The CPU is AMD Ryzen V1500B and package architecture is V1000 (x86+64).

Could someone please point me in the right direction?

As you wrote, it is x86+64, so you'd need to use the x64 version of Hard Disk Sentinel Linux:

From the Linux page https://www.hdsentinel.com/hard_disk_sentinel_linux.php you'll need to use

Hard Disk Sentinel 64-bit Linux console version - executable, zip-compressed
https://www.hdsentinel.com/hdslin/hdsentinel-020c-x64.zip

The PDF file uses exactly this, just an older version, as it mentions

https://www.hdsentinel.com/hdslin/hdsentinel-019c-x64.gz

(see bottom of page 11).
You can use that (it is still available for download) or use the newer 020c version.