Architecture
Before you YOLO deploy your Next.js app on a $3,99 VPS, it makes sense to take a few minutes and think about the overall architecture of your application.
Since you are not hosting on Vercel, you do loose all those nice benefits that come with it.
DNS & CDN
Maybe not relevant for your hobbiest side project, but for a "professional" project you should think about having some kind of DDos protection and/or a CDN caching your assets.
The commonly known providers (e.g. Cloudflare, Bunny, etc.) often do offer that in a bundle. With most providers a free tier is available.
Media Handling
Unless your whole applilcation contains just a hand full of images, you should probably not rely on the Next.js internal image optimization (based on sharp), to reduce the load on your server.
Using a custom image loader is the way to go here. an image and video optimization platform (check out the Next.js docs for recommended providers).
optional: Proxy
This really depends on your setup: if your are choosing the self hosted Docker route, you are going to need a proxy/app server routing the traffic to your Next.js application.
Popular choices are NGINX or Caddy. With Caddy being a little bit more end user friendly, since it does handle SSL/TLS termination for you. And NGINX being more performant, but a little heavier on the initial configuration.