$1000 doesn't really cut it anymore. If you want a graphics card that runs VR well, it's going to upwards of half of that budget. CPU + GPU + RAM can easily add up to $1000, you most likely will have to throw in a few hundred more for the motherboard + cooler + storage + PSU.
Yes, you can build a system on $1000. No, it will not meet your "smoothly and effectively" preferences at any decent resolution.
Recommended minimum parts:
GTX 1070 ti
i7-7700k
16 gb ram (this is more of a requirement, VR alone can use 8 gb and you need to have ram left over for everything else)
512 gb SSD
Any z270 motherboard
Edit: As stated in the previous replies, we can't really give you 100% accurate information without knowing the headset you want to use. As a side note, that alone can be $400 - $1000
An Occulus Rift S is about entry level as you can go as far as PC BASED VR is concerned. There are stand alone models that are cheaper. But they have significantly less power. There's some glitches with the Rift S however. (Just do a youtube of Rift S and see what they like and dislike)
The more pixels you have to pump, the meatier a system you will have to have.
I would say:
2070 Super ($500)
16 Gigs Ram ($100)
240 Gig SSD ($40)
CPU AMD 2600 $169
AMD B450 MB ($100)
case ($50)
500 watt PSU ($100)
~$1060 give or take for the PC Side
I would consider this entry level. You could go cheaper, but your system will quickly become dated. VR makers and programmers are quickly pumping the pixels up.