Iis – I can access site with host name IIS in local but not from other machines

When you enter a URL without a port, port 80 is implied.

No browser that I know of has any way to change this behavior; if you want to use port 3333, then you'll need it in the URL.

If I understand your question correctly, you want to do the following:

application-1.my-server.company.com point to my-server.cloudapp.azure.com:8080 
application-2.my-server.company.com point to my-server.cloudapp.azure.com:8081 
application-3.my-server.company.com point to my-server.cloudapp.azure.com:8082 

One solution is the use a reverse proxy. See this question for details: How do I redirect subdomains to a different port on the same server?. You can use self signed certificate between the reverse proxy and the web servers.

You also could offload the SSL certificate: https://kb.paessler.com/en/topic/44613-using-windows-iis-as-ssl-proxy-for-prtg. But then you need to manage multiple certificates. You could put a nginx reverse proxy before every application to add SSL support.

In you DNS add one A record for my-server.company.com. The best thing for the sub domains is to create CNAME records when they are on the same server. That way, you only have to update the A record when your IP changes. Otherwise create A records for every sub-domain.

For the SSL certificate. You have to have a certificate on every server. The traffic between your reverse proxy and web service has to be secure when you use multiple server. This because there could also be a man in the middle attack between your servers.