There is a good overview of what you can do here with guidance: https://docs.aws.amazon.com/lambda/latest/dg/monitoring-insights.html
There are a couple of different ways to do what you are looking for.
AWS Budget can be used to alert when you go over a threshold of spend OR usage. So you can set a threshold of 350,000 GB-seconds per month and it will tell you when you hit that limit. 
That won't tell you what Lambda is causing you the problem. For that, you can use CloudWatch with your Lambda Metrics to determine which Lambda is using the most resources.  (There is another answer here that talks about monitoring-insights)
- CloudWatch Alarms can also be used to alert on metrics.
Finally, you can also use Cost Anomaly Detector  to set an alert when an increase (or decrease) in usage/spend is detected. The threshold for the alert is set by the user, so you can set it for $1, $10, $100, $1000, whatever you want to know about.
Since you are talking about the Free Tier, it's important to note that the first 2 AWS Budgets are free, one CloudWatch Dashboard of up to 10 metrics is free and 10 CloudWatch Alarms are free. Cost Anomaly Detector is also free, but depending on your SNS settings for notifications, there may be a fee there. Therefore, it's possible to get the notifications that you are looking for with little or no cost.
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