If it is just the internal IP Address of your PC changing, due no doubt to PC and/or router reboots, which prevents the port forwarding from working in your router so that your connection to the PC from outside your network doesn't work, then you can just set up a DHCP Reservation for your PC in your router. A DHCP Reservation means that any time your PC connects to the router it will be given the same IP Address. Check the manual for your router. Most modern routers support reservations.
I don't know what you mean by "Change the IP address". Do you mean you change the IP Address that port forwarding points to? Or are you changing the PC IP Address itself, which could only be done if you are using a Static IP Address for your PC, assigned by your router? I recommend that you don't use static IP addresses on your LAN, and instead use DHCP Reservations as above. Your LAN will work much better and require less management, if any.
I don't understand how paying for an external Static IP Address with your ISP would help, if it is the internal IP Address that changes. But if the problem is that your external WAN ISP IP Address is changing, that is what the JRiver Access Key is for. Using the JRiver Access Key method to connect to your PC from outside your network gets around the problem of your external IP Address changing. JRiver keeps track of external IP Address changes and keeps it associated with your Access Key, so that any time you connect using your Access Key, the correct external Internet (WAN) IP Address is used.
Where the Access Key method may fail is if the PC was asleep and your external address changed, then river wouldn't know about it, so you couldn't connect.
So if it is your external IP Address that is changing, and you allow your PC to sleep, and your ISP changes your WAN Address fairly often, then you would need to connect from outside your network using the current external Internet address of your ISP connection. You can do that using one of the Dynamic DNS services available, if your router supports it. You don't say what you use to connect to your PC MC Server, but I assume it is JRemote, Panel, or Gizmo? I don't know if they support connecting using a URL provided by a Dynamic DNS service, but if they don't, you could use some other App to wake your PC remotely using the external IP Address provided by a Dynamic DNS service. Once the PC has been woken it will update the external IP Address associated with your Access Key, so then you can connect to your PC using the JRiver Access Key method.
I assume that your router supports Wake On LAN/WAN and you do have WOL/WOW working from outside your network?