Yep! You can pretty much deploy anything on DigitalOcean since it is a VPS. It acts pretty much like a dedicated server except just virtualized to have multiple people on a single system. Since you seem new here, you should probably have a look at the following to get you started:

1. Learn the Command Line
2. Getting Started with Linux
3. Ubuntu 16.04 Checklist
4. Install Node.js on Ubuntu 16.04
5. Running Node.js Apps on Ubuntu 16.04

Yes, you can deploy Node.js server.
DigitalOcean has great tutorials on setting up Node.js:
How to install Node.js on Ubuntu 16.04.
How to setup Node.js application for production on Ubuntu 16.04 is worth-look too.

There are also Node.js and MEAN stack One-Click images. It will setup Node (or whole MEAN stack in case for MEAN) on droplet creation so you can use it instant.
Only drawback is that One-Click images are based on Ubuntu 14.04 instead of latest Ubuntu 16.04.1.

Why not this steps are not automatically yet? 0.o
Access instances by ssh is insecurity and open possibilities for many errors.

Hello,

In addition to what has already been mentioned, here is also a quick video on how to deploy Node.js with PM2, Nginx as a reverse proxy, and Cloudflare for SSL termination:

Best,
Bobby