Having researched this on and off since asking, my conclusion is: "not that much is known".
The object was discovered on Sept. 3, 2002 by amateur astronomer Bill Yeung. The Paul Chodas from Center for Near-Earth Object Studies (CNEOS) published a news item on Sept. 11, tentatively identifying it as the S-IVB stage of Apollo 12:
Analysis of J002E3’s pre-capture orbit about the Sun shows that the object was always inside the Earth’s orbit, and that it may have come within the Earth’s vicinity in the early 1970s or late 1960s. Many of the test cases in our analysis in fact passed through the L1 portal, back into Earth orbit (going backwards in time) during the early 1970s. In other words, this object was very likely orbiting the Earth during this period before escaping into the heliocentric orbit from which it was captured in 2002. It seems quite likely that this object is one of the Apollo Saturn S-IVB third stages which flew by the Moon during this era (Apollos 8 through 12). The brightness of J002E3 seems to match the expected brightness of an S-IVB stage. Further circumstantial evidence suggests that this object is in fact the Apollo 12 stage, which was left in a very distant Earth orbit after it passed by the Moon on November 18, 1969. This spent rocket body was last seen in an Earth orbit with a period of 43 days, not much different from J002E3’s current orbit.
I was unable to locate any publication on this analysis other than this news item.
A follow-up from Dr. Chodas, published on Sept. 20 explains that further observations have improved orbit reconstruction and concludes:
Additional observations have now confirmed that the object was
indeed captured from a solar orbit earlier this year, and they
have also made clear that it escaped the Earth-Moon system in
March 1971. [...]
The timing of the object’s escape is consistent with our
theory that this object is the Apollo 12 S-IVB third stage,
which was left in a distant Earth orbit after it was launched
on November 14, 1969 and passed the Moon four days later. We
theorize that the spent rocket orbited the Earth chaotically for
15 months before finding the exit pathway through the L1 portal.
Again, no publication other than this news item.
A third update from Dr. Chodas from Oct. 10 concludes:
Looking into the past, we are still unable to connect the motion of
J002E3 with the last know position of the Apollo 12 S-IVB. One reason
is that the solar radiation pressure is not constant in time, but
rather changes with its position around the sun; to precisely account
for this effect we need to know the pole of rotation. Furthermore, if
J002E3 is the Apollo 12 S-IVB then that stage spent more than a year
in a highly chaotic orbit around the Earth. So far these two factors
have combined to prevent us from predicting the position of J002E3
with sufficient precision to definitively link these two
But again no publication associated as far as I can tell, also no further updates from CNEOS after this one.
In a 2012 publication (hat tip to Organic Marble), Daniel R. Adamo (who has a track record in state reconstruction, see e.g. this Q&A and references therein) reconstructed the trajectories of all the Apollo equipment that left for interplanetary space: "Earth Departure Trajectory Reconstruction of Apollo Program Components Undergoing Disposal in Interplanetary Space" (PDF). The section on Apollo 12's S-IVB reconstruction starts on page 40. The paper concludes, regarding J002E3:
Because the Apollo 12 S-IVB did not immediately depart Earth's vicinity due to a disposal targeting error, its fate is indeterminate with respect to this paper's scope. The trajectory reconstruction documented by this paper ends in a chaotic geocentric state with potential for collision with the Earth or Moon, along with the possibility of ejection into heliocentric orbit after a lunar encounter. This chaotic state is consistent with those observed in connection with object J002E3 circa late 2002, as documented in the report at http://neo.jpl.nasa.gov/news/news135.html
The CNEOS link is broken, but presumably pointed to the Sept. 20 update mentioned above (which was also captured by Spaceref.com) that also states that the orbit of J002E3 was chaotic and could not be definitely connected with the S-IVB trajectory.
The identification of J002E3 as a S-IVB stage was corroborated further by photometric analysis (reference, reference; both behind paywalls).
Concluding and answering the question:
- historic measurement data is presented in the paper by Daniel Adamo
- Somewhat recent trajectory data can be obtained from Horizons (search for J002E3; available time-span: March 1, 2003 to Jan 1, 2004)
- I was unable to locate any of the simulation data from the period from 1971 until 2003, nor was I able to find the raw observation data from September 2002 until J002E3's departure somewhere in 2003.