I have civic'01 car millage 85000km, It has some misbehaviors since last week, dash board lights blinking when AC turning on and meters down to zero and up again.It usually happens at low RPMs.
What could be the problem? Is it a problem of alternator or battery(Battery is also quite weak)?

When experiencing electrical problems, always start by checking the battery. Either remove the battery and place it on a charger or install a new battery (most likely).