How do I use retro drop down handlebars with no hoods?
You have the option of purchasing a product such as this:


These are secured to your drop bar where your brake hoods would normally reside. They provide a hand position on the ramp of the bar without having to use drop bar brake levers. They also can install directly over your tape (but it will eventually ruin the tape, so ideally you'd install them and re-tape everything.) While it would be the cheapest and easiest option, it doesn't address the issue of your hands being away from your brake levers (which is a very important feature when riding in traffic!) 

The existing brake levers are Tektro RL720s. They're known as "interrupter" levers because they are generally installed in tandem with drop bar brake levers and they interrupt the housing, creating two sets of brake levers at two different hand positions. You could purchase any aero levers and install them on the bar along with the 720s (they need to be aero levers where the housing runs under the handlebar tape). This would require some housing and new brake cables, but it would solve both the hand position and brake access issues you've posed for ~$40.