Can I configure a subdomain to point to a specific port on my server

If you have access to SRV Records, you can use them to get what you want :)

E.G

A Records

Name: mc1.domain.example
Value: <yourIP>

Name: mc2.domain.example
Value: <yourIP>

SRV Records

Name: _minecraft._tcp.mc1.domain.example
Priority: 5
Weight: 5
Port: 25565
Value: mc1.domain.example

Name: _minecraft._tcp.mc2.domain.example
Priority: 5
Weight: 5
Port: 25566
Value: mc2.domain.example

then in minecraft you can use

mc1.domain.example which will sign you into server 1 using port 25565

and

mc2.domain.example which will sign you into server 2 using port 25566

then on your router you can have it point 25565 and 25566 to the machine with both servers on and Voilà!

Source: This works for me running 2 minecraft servers on the same machine with ports 50500 and 50501

If you only got one IP on the server, there is no chance to do that. DNS is a simple name to number (IP) resolver. If you have two IPs on the server, you can point each subdomain to each of the IP-addresses and run both servers on the default port on each IP.
one.example.com -> 127.0.0.1 (server: 127.0.0.1:25565)
two.example.com -> 127.0.0.2 (server: 127.0.0.2:25565)

If you want to host multiple websites in a single server in different ports then, method mentioned by MRVDOG won't work. Because browser won't resolve SRV records and will always hit :80 port. For example if your requirement is:

site1.domain.example maps to domain.example:8080
site2.domain.example maps to domain.example:8081

Because often you want to fully utilize the server space you have bought. Then you can try the following:

Step 1: Install proxy server. I will use Nginx here.

apt-get install nginx

Step 2: Edit /etc/nginx/nginx.conf file to add the port mappings. To do so, add the following lines:

server {
    listen 80;
    server_name site1.domain.example;

    location / {
        proxy_pass http://localhost:8080;
    }
}

server {
    listen 80;
    server_name site2.domain.example;

    location / {
        proxy_pass http://localhost:8081;
    }
}

This does the magic. So the file will end up looking like following:

user www-data;
worker_processes auto;
pid /run/nginx.pid;
include /etc/nginx/modules-enabled/*.conf;

events {
    worker_connections 768;
    # multi_accept on;
}

http {

    ##
    # Basic Settings
    ##

    sendfile on;
    tcp_nopush on;
    tcp_nodelay on;
    keepalive_timeout 65;
    types_hash_max_size 2048;
    # server_tokens off;

    # server_names_hash_bucket_size 64;
    # server_name_in_redirect off;

    include /etc/nginx/mime.types;
    default_type application/octet-stream;

    ##
    # SSL Settings
    ##

    ssl_protocols TLSv1 TLSv1.1 TLSv1.2; # Dropping SSLv3, ref: POODLE
    ssl_prefer_server_ciphers on;

    ##
    # Logging Settings
    ##

    access_log /var/log/nginx/access.log;
    error_log /var/log/nginx/error.log;

    ##
    # Gzip Settings
    ##

    gzip on;

    # gzip_vary on;
    # gzip_proxied any;
    # gzip_comp_level 6;
    # gzip_buffers 16 8k;
    # gzip_http_version 1.1;
    # gzip_types text/plain text/css application/json application/javascript text/xml application/xml application/xml+rss text/javascript;

    ##
    # Virtual Host Configs
    ##
server {
    listen 80;
    server_name site1.domain.example;

    location / {
        proxy_pass http://localhost:8080;
    }
}

server {
    listen 80;
    server_name site2.domain.example;

    location / {
        proxy_pass http://localhost:8081;
    }
}

    include /etc/nginx/conf.d/*.conf;
    include /etc/nginx/sites-enabled/*;
}

#mail {
#   # See sample authentication script at:
#   # http://wiki.nginx.org/ImapAuthenticateWithApachePhpScript
#
#   # auth_http localhost/auth.php;
#   # pop3_capabilities "TOP" "USER";
#   # imap_capabilities "IMAP4rev1" "UIDPLUS";
#
#   server {
#       listen     localhost:110;
#       protocol   pop3;
#       proxy      on;
#   }
#
#   server {
#       listen     localhost:143;
#       protocol   imap;
#       proxy      on;
#   }
#}

Step 3: Start Nginx:

/etc/init.d/nginx start.

Whenever you make any changes to configuration, you need to restart Nginx:

/etc/init.d/nginx restart

Finally: Don't forget to add A records in your DNS configuration. All subdomains should point to domain. Like this:

Put your static IP instead of 111.11.111.111

Further details:

• http://tutorials.jenkov.com/nginx/index.html

• Redirect subdomain to port [nginx/flask]

• https://serverfault.com/questions/920534/nginx-proxy-subdomains-to-other-addresses-and-ports

Host Static Website: If you have any static website (Like angular app), that you want to deploy in Nginx itself. Place your index.html along with the other resources in some folder, like /srv/mySite and add the following server block in nginx.conf:

server {
    listen 80;
    server_name staticSite.domain.example;
    root /srv/mySite;
    location / {
       try_files $uri $uri/ /index.html;
    }
}