Chikmagalur town is small — you can walk the useful part of it in twenty minutes — and its relationship with coffee is stranger than you would expect. This is the district that grows some of India's best Arabica, and for decades the town itself mostly drank instant, because the good stuff was for export.
That has changed in the last ten years. Here is the honest state of it.
Places that roast their own
There are two or three at any given time, and which is best genuinely rotates depending on who is running the roaster that season. Ask whoever owns the estate you are staying on. They will know, they will have an opinion, and they will be right.
The filter coffee question
A proper South Indian filter coffee — decoction, chicory, hot milk, poured between tumbler and dabarah until it foams — is a different drink from an espresso, and it is what most people here actually drink. The small places near the bus stand do it best. Ask for it kadak if you want it strong.
Worth the drive?
From Kaapi Nest (16 km), yes — you can come in for dinner and coffee and be back within the hour.
From an estate 25 km+ out, probably not. The coffee on the estate came from forty metres away. Drive an hour for a worse cup and you have made a poor trade.