How do I set up a single subdomain to point to my droplet?

I have my domain at NameCheap (let’s assume it’s example.com) and my DNS is all managed by NameCheap and I really don’t want to change that. Now let’s assume I have a droplet at 1.2.3.4. What I’m trying to accomplish is point a single subdomain to my droplet, for instance, vps.example.com.

From what I could gather researching my problem I eventually created an A record on NameCheap’s DNS and pointed it to 1.2.3.4. But I’m assuming this is not enough because this IP is shared across many other droplets, right? Somehow it needs to know how to point to my droplet and not someone else’s.

I supposed that now I need to add the example.com domain to my droplet, correct? By default, I get the following DNS configuration on DigitalOcean:

• A @ 1.2.3.4
• NS ns1.digitalocean.com.
• NS ns2.digitalocean.com.
• NS ns3.digitalocean.com.

Now my questions…

1. Do I need the “@” record above if I simply want to have vps.example.com pointing to my droplet? Or can I simply replace the A record “@” with “vps”? Or maybe I need both?
2. Do I really need to have all the NS entries above or just some A record(s)?