On a budget, Hadrian's Library still earns its price.
€6 (summer) / €3 winter; included in €30 combo ticket
The surviving wall of Corinthian columns is striking and it's steps from the metro and the flea market, so it slots into a Monastiraki wander easily. The catch: it's a fairly small fenced ruin you can largely appreciate from outside the railings, and it doesn't merit a special trip on its own.