I'm sure your helmet would provide adequate protection in principle. Fundamentally, the job of a helmet is the same, whatever kind of bike you're riding. However, my suspicion is that this helmet won't work on a road bike because of the body posture while riding.
On a road bike, your back is much more horizontal than it is on a mountain bike, which means you must lift your head to see forwards. However, the visor of your helmet is likely to block your forward view. It's probably removable, in which case this wouldn't be an issue. What looks like a bigger issue to me is that your helmet comes a long way down the back of your head. This would make it difficult to lift your head to see forwards.
You also say that the helmet is "old" so, honestly, it's probably a good idea to buy a new one anyway. The materials degrade over time so old helmets offer less protection.