I am looking to assign a static IP to my Lambda which is being invoked via the API gateway. This is required because, the downstream system that i invoke from this lambda accept
You will need to:
The NAT Gateway will use an Elastic IP address (which is a static IP address). All traffic from the Lambda function to the Internet will come from this IP address, which can be used in the whitelist.
You might think that this is a bit of overkill for simply attaching a static IP address, but multiple Lambda function can run in parallel and they could run in multiple Availability Zones. Sending all traffic through the NAT Gateway is the only way to ensure they all have the same IP address. (Or, to be more specific, one IP address per AZ in which the NAT Gateway is launched.)
You can't assign a public/static IP to any Lambda function.
Your only good option is to deploy into a VPC with an Internet Gateway and configure routing from the Lambda's subnet through a NAT which has an Elastic IP. Then your target host can whitelist the Elastic IP.
Also see:
The way to assign a static IP address without a VPC is AWS Network balancer.
I agree with the answer by John for having static IP whitelisting part. However, it won't resolve your cold start problem because lambda,if ideal, actually takes a small time to start. So I would recommend you also create a Cloudwatch event to hit lambda periodically to resolve this or write a simple code(either in lambda or somewhere else) which sends an empty request periodically so that cold start problem is resolved. You can view the improvement in X-Ray. This is an overhead but one time process.