Here you need to apply the -r (or --role-name) option on to cdk itself (and not to the deploy option):

cdk --role-arn arn:aws:iam::1234567890:role/MY_ROLE_NAME deploy

cdk will generate a Cloud Formation template and deploy it using the role you specified.

You configure the Lambda Function's environment variables with its Environment prop as a *map[string]*string.

You'd typically pass the values down as stack props from its parent stack to the Lambda construct. Or simply hardcode the values in the Function construct code. The CDK best practice is to have such input values fixed at synth-time, so that deploys are deterministic and under source control.