Can I connect my Time Capsule wirelessly to my mobile router? I've just received a router with my new internet access. This router is wireless only and therefore has no ports except for power.

How can I connect my Time Capsule to this router now that I can no longer connect it directly through a network cable?

Using the AirPort Utility, I've tried to choose the Wireless Network Mode: Extend a wireless network. Even though I provided the correct settings for the existing wireless network my Time Capsule doesn't extend it and there's no wireless connection to my wireless router. It just keeps blinking orange...
I'm afraid you can't.
Extending networks only works between Apple products and the Time Capsule does not have a "client mode", as the Airport Express. You could buy a router that can connect wirelessly to your ISP Modem - Router, and connect the TC by an ethernet cable to it, in order to keep using it as a Time Machine destination. You would have to put the TC in bridge mode. An Airport Express will be able to fulfill this role.