The grounds of the Imperial Palace used to be swarming with samurai in shining armour, ready to fend off an attack to the Shogun, who resided in the castle. Nowadays, some 140 years later, the swift samurai have been replaced with swift Tokyoites in flashy running shoes (mind you, running around Tokyo requires the hippest of running wear).
The 5km course has been equipped with lockers , showers, toilets and fresh water fountains. Furthermore, there are many palace guards lining the course, so running at night is perfectly safe even if you are by yourself, although the course is so popular that you will probably never really be by yourself. 
Photo courtesy of rocket9.net
Here are some shower opportunities around the course:
Bain Douche Public Bath
Located just around the corner of the Imperial Palace, or a 1-minute walk (or a 15 seconds run) from Tokyo Metro’s Hanzomon station (exit 2).
Address: 1-5-4, Koujimachi, Chiyoda-ku
Hours: 15:00–0:00 (Sat and regular holidays: 13:00–23:00)
Price: around 400–500 yen
Inariyu
A public bath located in Uchikanda and close to both Otemachi and Marunouchi station. This bath was used as a location for the immensely popular Japanese movie “Thermae Romae” about a Roman man who time-warps to Japanese bathhouses. Good for an after-run freshing up. There is also a Jacuzzi.
Address: 1-7-3, Uchikanda, Chiyoda-ku, Tokyo
Hours: 14:50-01:30 (regular holidays: 14:50– 22:30)
Runners Station Jinbocho
This runners' station provides not only showers, but also lockers. The showers come with amenities such as shampoo, so that saves space in your running bag. A two-minute walk from Tokyo Metro’s Kudanshita station, and just 500m from the Imperial Palace.
Address: Yasuda Jinbocho Mansion 1st floor, 3-11-1, Chiyoda-ku, Tokyo
Hours: 7:00 – 22:00 (Sat: 8:00 – 22:00, regular holidays: 8:00–18:00)
Runners Station Plus Bike Koujimachi
This station is located next to exit 4 of Tokyo Metro’s Koujimachi station. It features seven showers for each gender and 148 lockers – 78 for males, 70 for females. They provide the usual soap, shampoo and conditioner so you don’t have to bring it. A bicycle rack is available, so this station is used by many ‘imperial’ cyclists too.
Address: Koujimachi Crystal City building 1st floor, 4-8-1, Koujimachi, Chiyoda-ku, Tokyo
Hours: Mon-Fri 7:00 – 22:30 (Sat 8:00 – 20:00, Sun and regular holidays 8:00–18:00)
JOGLIS powered by TOKYO FM x asics
This station is located underneath a radio station, therefore the name is a conjunction of ‘jogging’ and ‘listen’. JOGLIS is a 2-minute walk from Hanzomon Metro line exits 1 or 2, and is located across the street from the palace. It also features a runner’s shop, might you forget your running shoes or your shirt.
Address: Basement level 1, 1-7 Koujimachi, Chiyoda-ku, Tokyo
Hours: Mon-Fri 7:00 – 22:30 (Sat 7:00 – 19:00, Sun and regular holidays 7:00–18:00)
Website: http://www.joglis.jp/index.php in Japanese