Step 1: Understand the phrasal verb.
The phrasal verb keep up means to maintain, preserve, or continue something at the same level or standard.
Step 2: Interpret the sentence.
The sentence states that the person keeps up the reputation of the institution wherever he works. This means that he consistently maintains the institution's good name and standards.
Step 3: Analyse the options.
• (A) deserves -- Incorrect. It means to be worthy of something, which is unrelated.
• (B) maintains -- Correct. It exactly matches the meaning of keep up.
• (C) appreciates -- Incorrect. It means values or admires, not maintains.
• (D) owns -- Incorrect. Owning something is different from maintaining it.
Step 4: Final conclusion.
\[
\boxed{\text{Keep up}=\text{maintain or continue}}
\]
Hence, the correct option is \(\boxed{(B)}\).