Not able to connect to internet from Amazon EC2 instance linux box

In the amazon EC2 instance, i have set the set the inbound/outbound of security group to MY-IP address.

When i try to install the mysql client or some other software, its saying connection timeout.

But when i make the inbound/outbound to anywhere ie 0.0.0.0/00, then i am able to install the software.

But i don't want to expose the EC2 instance requests to everyone.

How to connect to Internet and install the software with restricted security groups. Please can u help out on this.