Is it possible to perform update on the fetched data from external system without SAVING the same in salesforce and sending the updates back to the external system by using web service callout?
Yes, so long as you can stay within the Callout Limits and Limitations.
Generally that means making fewer than 100 callouts and staying within the 120 second cumulative callout timeout.
So within one Apex transaction you can retrieve data via a callout, modify it, and then make another callout to update it.
You would also need to stay within the Apex CPU and heap size limits.