We did a little digging around and found a helpful article (in Chinese) on nongjiayuan.org (a Chinese language website that lists fruit orchards where visitors can go picking around town).
Here is a rough translation of the information on this site pertaining to the Sijiqing Cherry Picking Festival at Yuxiang Yuan Park:
“Dates: May 22 to [around]June 10 (note: this date is an approximation, according to the website, the actual end date is tbd, according to to the park operator)
Picking Price: pick of this year is still 100 yuan / Zhang, two jin can be taken away with the price of admission, anything in excess will be charged at 50 yuan / kg.
Driving Directions: Take the Fourth Ring Road to the Yuquan Shan exit (玉泉山出口), go west on Minzhuang Lu (上闵庄路向西) until you reach the station for bus route 45 (到四十五中车站就可以看到园子了) – the orchard should be visible from there.”
Tel: 6259 0135 (Chinese only)
Check out the rest of the site for more information about other orchards where you can go picking around town.