I don't know of any solutions that are super simple, so I will tackle this issue from an shopify app developer perspective.
From what I understand, you have an MySql database that you keep to updated separately from some separate software. When you update that database you want to update shopify to reflect the changes to products.
I don't know of any app that does this so this probably consists of custom code. This situation is a bit tricky, because it depends on your that separate software that you use to manage that products.
Building a custom app:
Polling: A custom app on shopify will connect to your mysql database every so often to query for any changes that you've made over the last while. Then it will fire API calls to shopify to update the values in Shopify. There are potential issues and time delay related to updates through polling.
Event Driven: This is where you would need your separate product management software to fire an event to the custom shopify app to update Shopify's data. This is the better approach, but can be more difficult to implement or impossible depending on your Product Management Software.
Both of these require some background knowledge and information about how the shopify app/API system works and programming in general.
Best of luck.
If you need professional assistance, feel free to contact me/us.
Yup, we might be oversimplifying some of this, but yup through a cron job.