I am working on Azure Poc . Is it possible to connect to on premise Sybase database from Azure cloud ?
Can I read and write data into Sybase from Azure cloud ?
Yes you can. Look at Azure Relay Hybrid Connections for a code-first approach or Site-to-Site VPN for an infrastructure one. Also since you're on a Linux VM, why not just SSH tunnel your traffic over? - probably the simplest solution to your conundrum.
There's also ExpressRoute which you may want to take a look at if we're talking serious business that needs connectivity, bandwidth and latency guarantees.