Yes. It is possible. A JTAG header on the board can connect with the USB Blaster.
Regarding power supply, the Sensor board can be used as a module that takes power from USB, and the MAX10 board alone can also be powered from an external pin using a breadboard or the like.
See the "Tips and Tricks" section of the User Guide for a list of parts that can be used for powering.

Odyssey User Guide (pdf 2.8MB)

If you have a Sensor board, you can use the preloaded firmware to program FPGA files without a USB Blaster.
Normally you would program the .sof file through the JTAG header using a USB Blaster, but in this case you would convert the sof file to a JBC file, load it into a general Personality slot (menu option 5), then program the FPGA with menu option 6. To do.

If you want to try a new FPGA image, you must first clear the Personality slot (menu option 2) and then load the personality again (option 1).
The development process is a bit more work, but we provide a way for those who don't have a USB Blaster to program it.