While I cannot guarantee this answer, using the help of Rikaichan, it appears there are slight differences between each word. The correct answer makes sense, if I read each of them as I assume they are to be read.
回復 is often used for general purpose healing. Think of a person recovering from a sickness. 回復 is often used for healing in video games, if that helps you.
修復 means "restoration, repair, or mending." If I am to understand the given sentence, it is discussing the repair, mending, fixing, etc. of a picture from over 300 years ago.
復旧 means "restoration, restitution, or rehabilitation." This would likely be my second choice, but "restitution" and "rehabilitation" make me believe this is more likely to be used in other contexts.
復興 means "revival, renaissance, reconstruction, or restoration." I imagine this is more of an abstract concept. Think of time periods where lots of destruction occurs, and life slowly regains its natural state. That is more so the "revival" or "restoration" I would think.
Of course, this is my pure conjecture, and I do not believe I could even pass JLPT2, but this question seemed to make sense to me. Take it as you will!