Azerbaijan's Ruslan Lunev won the gold, while Turkey's Murat Kilic claimed the silver.
Hashmi had earlier finished third with a total score of 568 (286 and 282 in stages one and two) in the qualification round to advance to the final along with his teammate Hilal al Khatri. Khatri shot a total of 546 (272, 274) to also advance with a sixth-place finish. In the final, Khatri was eliminated in the first stage.
Another Omani shooter, Mohammed al Gheilani, shot a total of 537 to make an early exit in the qualifying round.
Except for Hashmi's medal, there was nothing else to cheer for the Oman camp as shooter Wadha al Balushi finished eighth in the final of the 25m women's pistol event.
In the qualification round, Wadha was seventh with a total score of 560 after rounds of 286 (95, 94, 97) in precision and 274 (89, 95, 90) in rapid to advance. Her teammate, Sahir al Jahmani, shot a total of 537 to finish 12th among 17 competitors in the qualifying round.
Qatar's Aldana Saad Almubarak won the gold, while Iran's Haniyeh Rostamiyan took the silver. Three-time African champion Olfa Cherni of Tunisia managed to clinch the bronze medal.
In mixed team 10m air rifle event, Oman's Siham al Hasani and Abdullah al Habsi shot a combined score of 507.6 to qualify for the final, but could finish eighth while the other duo of Mohammed al Hajri and Amina al Tarshi shot a combined score of 506.4 to exit in the qualifying phase after finishing tenth among 12 teams.
Meanwhile, the national football team, coached by Hamad al Azzani, will take on Algeria to decide the topper in Group B on Monday.
Oman drew its opener against Palestine 2-2 and then beat Turkey 3-2 in the next to collect four points, while Algeria won both its games to currently lead the standings with six points.