This is not meant to be a gotcha question but the use of free books online. It is interesting that, like everything else, "time" was kept differently back then than now. If you didn't already know the answer, hopefully it's a bit of trivia you enjoyed learning.
According "The Portal to Texas History" chapter, a Time Table in the Texas Almanac, in Austin, TX at 12:00 noon 1863 it would be 1:23 pm in Washington DC.
However, World-TimeDate.com, at 12:00 noon in Austin, Texas in 1863, it would have been 01300 or 1:00 PM in Washington D.C., The actual times zones were not created until 1883, and changed over time to what we use today. During the Civil War there were hundreds of "local time zones".
109 degrees 02' 13" Washington DC
- 97 degrees 44' 35" Austin, TX
11 degrees 58' 13" round off 12 degree
4 minutes of time per degree
4 x 12 degrees = 48 minutes
Austin, TX Noon
Washington, DC 12:48 pm
I don't have the almanac handy, but Austin is 20deg 47.5min west of DC. If each 15deg of longitude represents one hour of the Earth's 24-hour rotation (360deg/24 =15deg), that should put Washington ahead by 1.39 hours, or 83 minutes. So my answer is, 1:23 p.m.