Yes, you need to check the presence of a voltage combined with the absence of a current.
Google current sensor ICs. You can get current sensing modules from Banggood et al, which are handy because the components are already mounted on a PCB with a convenient connector.
Each circuit you are sensing will need two ports on the Arduino - one for the voltage, the other for the current. In your code you will run a loop which scans each of the voltage pins in turn, round and round. When a voltage is detected, the software should then check for an output from the associated current sensor. If it is absent, or the wrong value, then sound an alarm.
That's the high level view - there is much detail to be added in the implementation.
When you've got that working, you could than develop the Mk2, which continuously monitors the bulbs by feeding them a very small voltage (not enough to light the bulb) and detecting the small resultant current, which will disappear when the bulb fails.