You need to start by removing the backglass. Whoever sold it to you, should have given you a key to unlock the backglass. Once unlocked you need to slide the glass up to remove it. This site has a lot of folks to help you, but pinball machines are very complicated to work on. Be prepared to be overwhelmed but be patient and you might be able to fix it without hiring a pin repair man.

Side note, I can almost 100% guarantee your issue has nothing to do with dead batteries. It may have to do with really old batteries that leaked gas and damaged the main board. I hope that isn’t the case.

Once you open up the back and get to the boards, read this Link below and start to diagnose the boot cycles and where it is stopping. You have a system 11c.

http://techniek.flipperwinkel.nl/wms11/index2.htm#cpuled

You definitely have an issue with tripe main board. I am not good at diagnosing these things via Pinside threads. There are many others here that are.

Did you look and see if the batteries are leaking. If the board is damaged from batteries, you may be in bad shape.

Here is your worst case scenario. Replacement board.

https://ksarcade.net/rottendog-mpu011c-for-williams-system-11c.html

Check power supply voltages and also look for blown fuses on it.

Agreed check the power supply and fuses, if the power is good, focus on troubleshooting the main mpu/driver board.

Do you have and can you use a digital multimeter? Check the power supply test points. Or at the power in connector on the cpu. Batteries are in the middle of the main board if it has them still. Hopfully its had them removed and nvram added.
Can always post a picture of the boards and it may tell the tale.
Should be something simple unless there is prior acid damage or current ongoing acid damage.