Can you use a 12V 150mA adapter on a device that requires DC 12 V at 1 A?

As long as the current output is greater than power supply it replaces it will work fine. If the replacement is rated for less current (amperes) than the original you don't want to use it.

Of course a DC supply has to be replaced with a DC power supply. AC with an AC supply. The polarity should be the same and the voltage should be the same and the pin and barrel should be the same diameter as well.