Take a bus from CMBT, Take 15B bus. It starts almost every 5 mins. So, you will have time to keep you luggage as well as get a seat. It will take 40 Mins from CMBT to Thina Thanthi office which is the back entrance for Egmore.
If you don't want bus Auto/Call Taxi is the best option.
PS: Once metro becomes operational in future you can take metro and can reach Egmore/Central in 20-30 Mins.