When John Carter is returned to Jarsoom/Earth by the Thurns, he arrives in his body in the gold cave and is suffering from atrophy of his muscles. The sheriff is also just a pile of bones.
How much time passed on Earth since his few days on Mars?
We are not given a frame of reference beyond the decayed and skeletal remains of the Sheriff. Left with that we are forced to use science to determine how long it may have been. For that we turn to a paper from the University of Tennessee discussing "Decay rates of human remains in a arid environment" conducted by Galloway A, Birkby WH, Jones AM, Henry TE, Parks BO.
From their abstract of the paper:
The environment of southern Arizona with mild winters and hot, dry summers produces great variability in decay rates of human remains. Summer temperatures, which range well over 38 degrees C (100 degrees F), induce rapid bloating as a result of the accumulation of decompositional gases. However, in certain circumstances, the aridity can lead to extensive mummification, allowing preservation of remains for hundreds of years.
A retrospective study of 189 cases, concentrating on remains found on the desert floor or in the surrounding mountains and on remains found within closed structures, outlines the time frame and sequences of the decay process. Remains can retain a fresh appearance for a considerable time in the winter, but the onset of marked decomposition is rapid in the summer months. Bloating of the body usually is present two to seven days following death. Following this, within structures, there is frequently rapid decomposition and skeletonization. With outdoor exposure, remains are more likely to pass through a long period of dehydration of outer tissues, mummification, and reduction of desiccated tissue.
Exposure of large portions of the skeleton usually does not occur until four to six months after death. Bleaching and exfoliation of bone--the beginning stages of destruction of the skeletal elements--begins at about nine months' exposure. Insect activity, including that of maggot and beetle varieties, may accelerate decomposition, but this process is greatly affected by location of the body, seasonal weather, and accessibility of the soft tissues. Carnivores and other scavengers also are contributing factors, as are clothing or covering of the body, substrate, elevation, and latitude.
Given this information: We can assume that the skeleton, with active insects or other predation could have been there for as little as two months or as long as six months. There appeared to be profound weathering and the conditions were appropriately hostile as well, potentially speeding the process. Wherever or whenever John Carter is, time is moving at a compressed rate in comparison to his time on Earth.
Considering both of these answers I have two theories:
If he had left Earth for three years and only spent a week on Mars then he would have only been away from Mars for 30 weeks, give or take, after the therns sent him back.
If Po was dead for four months then John was away from Mars given still that only a week at most passed is still about 30 weeks
lol both work and both sit the same time frame.