If his age is not given in the information I posted, then, it is not available in the records I have access to.

I suspect that if he died on the Gettysburg battlefield, then he would have initially been buried in a mass grave there. Apparently there are few Confederates buried in the National Cemetery there, see:

A few Confederate soldiers were mistakenly interred in the cemetery alongside their
former foes. Although they fell under different flags, they now rest under one. However,
in the bitterness sown by the war, the remainder of the Confederate dead remained buried
on the battlefield. Between seven to ten years after the battle, these Southern remains,
over 3,200 altogether, were returned home to four primary locations – Richmond,
Virginia; Raleigh, North Carolina; Savannah, Georgia; and Charleston, South Carolina.
http://www.nps.gov/gett/planyourvisit/upload/National%20Cemetery%20brochure.pdf

Here is information on four J. W. or James Kirks in Mississippi units. Perhaps their regimental history sketches might narrow your search:

James Kirk, Private, Company A, 29th Mississippi Infantry, enlisted April 28, 1862 at Grenada, Mississippi, last recorded present on the March & April, 1864, rank of 4th Sergeant, no further records

29th Regiment, Mississippi Infantry

29th Infantry Regiment, organized at Corinth, Mississippi, in April, 1862, contained men from Grenada, Lafayette, Panola, Yalobusha, Washington, and De Soto counties. The unit served in Mississippi, then moved to Kentucky where it saw action in Munfordville. Later it joined the Army of Tennessee and was placed in General Walthall's and Brantly's Brigade where it participated in many battles from Murfreesboro to Bentonville. The 29th lost 5 killed and 36 wounded at Munfordville, had 34 killed and 202 wounded at Murfreesboro, and suffered fifty-three percent disabled of the 364 engaged at Chickamauga. It reported 191 casualties at Chattanooga and in December, 1863 was consolidated with the 30th and 34th Regiment and totalled 554 men and 339 arms. This unit reported 5 killed and 22 wounded at Resaca, and in the fight at Ezra Church the 29th/30th lost 8 killed and 20 wounded. Very few surrendered in North Carolina in April, 1865. The field officers were Colonels William F. Brantly and Edward C. Walthall, Lieutenant Colonel James B. Morgan, and Majors Newton A. Isom and George W. Reynolds.

...............

James Kirk, Private, Company E, 24th Mississippi Infantry, captured at Chattanooga, Tenn., November 24, 1863, forwarded to Provost Marshal, Louisville, Ky. November 30, no other records

24th Regiment, Mississippi Infantry

24th Infantry Regiment was organized at Meridian, Mississippi, during the fall of 1861 with eleven companies. Its members were recruited in the counties of Hancock, Clay, Lowndes, Chickasaw, Kemper, Choctaw, and Monroe. It moved to Florida, returned to Mississippi, and took part in the siege of Corinth. After serving in Kentucky, it was assigned to General Walthall's and Brantly's Brigade, Army of Tennessee. The 24th participated in many battles from Murfreesboro to Atlanta, endured Hood's winter operations in Tennessee, and saw action in North Carolina. It sustained 116 casualties at Murfreesboro, 132 at Chickamauga, and 189 at Chattanooga. For a time it was consolidated with the 27th Regiment and in December, 1863, totalled 491 men and 354 arms. At Resaca the unit lost 24 killed and 28 wounded, and at Ezra Church the 24th/27th had 11 killed and 67 wounded and missing of the 430 engaged. About 25 men were present at the surrender. The field officers were Colonels William F. Dowd, Robert P. McKelvaie, and R.W. Williamson; Lieutenant Colonels Clifton Dancy and William L. Lyles; and Majors George M. Govan and William C. Staples.

.............

J. W. Kirk, Private Co. E, 12th Miss. Cavalry, enlisted November 20, 1863 at Lexington, Mississippi, recorded on the July & Aug, 1864 at absent, "Sent by order for the defense of Mobile," listed at the end of 1864 as absent without leave, no further records

12th Regiment, Mississippi Cavalry

12th Cavalry Regiment [also called 16th Confederate Cavalry] was organized during the summer of 1863. Many of the officers and men were members of the Mississippi state forces and some were from Louisiana. The unit was assigned to Richardson's, Ferguson's, and C.G. Armistead's Brigade, and fought in various conflicts in Mississippi and Alabama. Many were captured at Selma, but the unit was included in the surrender of the Department of Alabama, Mississippi, and East Louisiana. The field officers were Colonel C.G. Armistead, Lieutenant Colonel Philip B. Spence, and Major William Yerger, Jr.

............

J. W. Kirk, Private, (1st) Company B, Yazoo Battalion Minute Men, Mississippi State Troops*, drafted June 21, 1862 in Yazoo County, last recorded present on the February 28, 1863 muster roll, no further records

* This company subsequently became (2nd) Company C, 2nd Battalion Mississippi State Troops
[Note: I cannot locate a history of this unit]