Most likely the display technology on your old monitor is not up to the task and you will need a new monitor, OR an adapter, that being said smaller HDMI monitor are not that expensive. You will need you get a monitor that supports HDMI or Thunderbolt. Or you can use some sort of adapter, BUT you will lose much of the resolution that the mac can support.
Here's an example of one that would work....
Amazon.com: Asus VE228H 21.5-Inches LCD Monitor: Computers & Accessories
This is the possible adapter you need (or you might need a Thunderbolt to DVI adapter).....
http://www.amazon.com/AmazonBasics-M...mUvbUpU2632792