Will 9v 1000mA power adapter work with your 9v 300mA device?

Yes you can, however in an unusual situation (i.e. electrical fault in the device), the adapter will allow 5x the current the device is rated for to pass into that device. In normal operation, the device will draw what it needs, based on fans running or not, etc. Under these conditions, as long as the adapter puts out the proper Voltage, the device will operate just fine.