I started this as a comment, but ran out of room. It doesn't actually answer the question, but explains why the question can't be answered in a meaningful way.
I'm sure you're aware that Jan 1 isn't the time the equinox happens, but the J2000 reference frame is based on the Earth's orientation at that time. By the time the actual equinox comes around the Earth will have precessed by then, and where the Sun actually crosses the equator, won't be the J2000 equinox.
So, honestly, it's better to just accept that the x-axis is where it is "by definition". It's a computed value rather than an actually observed one. The computation only includes precession, not nutation, polar motion, nor delta T. So, even if Jan 1 was the equinox, the Sun still wouldn't be at exactly 0,0 RA/Dec when it crossed the equator.
In fact, the ICRS has replaced the J2000 system. It has been aligned to the J2000 system to within a very tight margin. But it is not defined by the equinox, instead there are very distant radio sources which are assigned RA/Dec values, and those coordinates are correct by definition. Whatever happens to the Earth isn't relevant.
If you still want to know where the 0,0 RA/Dec point was on Jan 1, 2000, this can be done by computing the Geographic Position of 0,0 at Jan 1, 2000 12:00:00 UT. The geographic position of any object, at any time can be computed by:
\theta &= \delta \\
\phi &= \alpha - GST
$ \theta $: Latitude
$ \phi $: Longitude
GST: Greenwich Sidereal Time (Mean or Apparent depending on accuracy needs)
$ \alpha $: Right Ascension
$ \delta $: Declination
Since we're looking for the 0,0 point, $\theta=0$ and $\phi=0$. GMST at that time was about 18.697374, which equates to a lat/lon of:
+00° 00' 00.00" N
+079° 32' 21.76" E
Here is a page which computes the geographic position of the Sun and many stars for any time. It uses Mean sidereal time rather than apparent, so the actual position will be off by a bit.