I have an Amiga 4000 also. It's not really going to support a true VGA monitor without a graphics card. You would need a monitor that supports 15.75 khz to even set the up one of the VGA screenmodes. Your best bet to test it out is an old A520 composite adaptor from an Amiga 500. Otherwise to really get the most from your 4000 you will need an NEC 3D, C=1942/1960, or other monitor that supports both 15.75 and 31+ khz screens. Fire it up without a monitor and see if you get any HD activity. You'll be able to tell without a monitor by watching the power and hard drive leds if it's booting up.
If it works I would STRONGLY recommend a Picasso IV instead of a special monitor. Alternatives might be some other flicker fixer/scan doubler and graphics card combo. Perhaps an old GVP Spectrum or Picasso II and an Amber board A2320? This way you can use virtually any monitor instead of being locked into older specialty ones. I LOVE the Picasso IV in my A4000.
BTW open that puppy up and make sure the battery isn't corroding away the motherboard!
Congrats!
Jeff