How to bypass closed ports in local network to play games

tl;dr: ports to play games blocked on local network, want to bypass them and play again

I live in a uni dorm with a pretty good internet connection (1 gbps), the only thing is that the only open ports are those of the following protocols:

  • web (HTTP, HTTPS, passive FTP, Squid, Webcache)
  • mail (POP3, POP3s, IMAP4, IMAPs, SMTPs and SMTP submission via STARTTLS)
  • secure protocols (SSH, IPsec VPN, OpenVPN, PPTP, RDP)

This means that without a VPN (free ones are slow and being a uni student I'm pretty broke atm) every program or game that requires any other ports does not work, such as Battle.net, League of Legends, CSGO and so on, so I'd like to know what's the best way to tunnel those required ports from my computer to a remote computer (like an Amazon EC2 instance or something) and vice versa. I tried making the ec2 instance a proxy with tinyproxy and squid following tutorials but despite every port on the instance was open it didn't work at all (I'm not a sysadmin so I may have missed something with the configuration), and I also tried doing ssh -D <ec2-instance.com> user@<ec2-instance.com> -i <keyfile.pem> on port 8080 (open on local network) and used it as a proxy, and it only worked on the ports I didn't need such as port 80 and 443 but no way to connect to Battle.net or League and so on (iptables on instance were completely clear and all traffic was allowed, no trace of firewall on the instance), so please tell me the best and right way (and how) to do such a thing so that games can connect to the ec2 with open ports to be forwarded to me and the other way around (it has to be able to forward both TCP and UDP, this is very important). As I said I do have access to something to connect to that has unlimited traffic. Many thanks in advance.

P.s.: sysadmin told me the network is already configured (he can't forward them for me) and to use a VPN instead.