The Express will join conventional WiFi networks that use WPA2 Personal or WPA/WPA2 Personal security, but a captive portal probably requires a special type of authentication called Enterprise.
Unfortunately, Apple's setup "wizard" will not present Enterprise authentication options when you try to set up the Express using the standard methods
What operating system are you running on your Mac? We ask because there may be a way to manually set up the Express to see if it might work. I don't have a captive portal here to test the setup, nor have I tried this in the past.
You mentioned that you wanted the AirPort Express to "create" a home network. It cannot do this if it is set up to join a 3rd party network. With great luck, the Express will provide an Ethernet signal to the Time Capsule, and it is the Time Capsule that will provide your wireless network.
It is not technically correct to say that the Express will function as your modem in this type of setup, but in a way this is similar to the type of service that it will provide.
There are no guarantees that any of this work, but we can tell you how to try if we have the information about what operating system your Mac is running, or if you are using AirPort Utility on a PC.....and you have the time to experiment with low expectations about success.