How can I arrive at the U.S. on an E-3 visa then turn it into an EB-5?

To enter on any visa other than H-1B or L-1, you need to have an unrelinquished residence abroad. So if you have no intention to stay upon entering as an E-3, it would be best to immigrant visa process your EB-5 abroad at the US consulate in your home country.

You can file Form I-526 with Form I-485 concurrently to adjust status in the U.S. if you are in the U.S. in valid E-3 status. Once the application is filed, you would not be able to depart the U.S. and return in E-3 status until your advance parole travel permit is issued - doing so would cause your I-485 application to be abandoned.

The less you talk about your immigrant intent, the better. You cannot apply for an EB-5 *visa* in the US; you can file the EB-5 petition + adjustment of status to residence. Most of my E-3 clients choose to file the EB-5 petition and then consular process the visa so they may freely travel in/out of the US while the EB-5 is processing.

That should not be a problem - there is an issue of nonimmigrant intent once you file for your green card so I would not travel until you get the travel permit (advance parole) which you can obtain as an interim benefit with the I-485 filing.

If you enter on the E-3, 90 days after entry you can change your intent. Therefore, you can file the EB-5 petition and adjustment of status (permanent residence application) concurrently.