Is it possible for me to somehow upload the application I have written with C++ in Atmel Studio with the Arduino ISP that I can use in the Arduino IDE?
I have no idea how to do this, and I also have no idea where to set the fuses and what to set them to (I want to use the 8 MHz internal clock).
You want to use AVRDUDE. It will allow you to specify files for writing and verifying flash and EEPROM, as well as values for fuses. You will need to use a number of command line arguments, an example of which is given at Adafruit's ArduinoISP tutorial. Simply replace the partno argument with t45 to target the ATtiny45.
As for the fuses, use the Engbedded Atmel AVR® Fuse Calculator to figure them out. Pick a device, enter the current settings into the entry boxes at the bottom, change the desired options in the configuration boxes above, and write the new fuse values to the chip.
