There are three options to get from the Hongu area (Yunomine Onsen, Kawayu Onsen) to Tokyo.

By bus, train via Kii-Tanabe and Shin-Osaka

Time: ~8 hours (depends on connection times)

Route Explanation Tokyo to Hongu (Yunomine Onsen, Kawayu Onsen):

This route requires two train and one bus ride.

[Shinkansen] Tokyo to Shin-Osaka
[Train] Shin-Osaka to Kii-Tanabe (Kuroshio) (~4750 yen, non-reserved seat)
[Bus] Kii-Tanabe to Hongu area (bus timetable 1) (~2000 yen)

Route Explanation Hongu (Yunomine Onsen, Kawayu Onsen) to Tokyo:

This route requires one bus and two train rides.

[Bus] Hongu area to Kii-Tanabe (bus timetable 2) (~2000 yen)
[Train] Kii-Tanabe to Shin-Osaka (Kuroshio) (~4750 yen, non-reserved seat)
[Shinkansen] Shin-Osaka to Tokyo

By bus, train via Shingu and Nagoya

Time: ~7 hrs (depends on connection times)

Route Explanation Tokyo to Hongu (Yunomine Onsen, Kawayu Onsen):

This route requires two trains and one bus ride.

[Shinkansen] Tokyo to Nagoya
[Train] Nagoya to Shingu (Wide View Nanki) (~6870 yen, non-reserved seat)
[Bus] Shingu to Hongu area (bus timetable 3) (~1540 yen)

※The last bus from Shingu to the Hongu area which goes via Kawayu Onsen and Yunomine Onsen departs at 15:55.

Route Explanation Hongu (Yunomine Onsen, Kawayu Onsen) to Tokyo:

This route requires one bus ride and two train rides.

[Bus] Hongu area to Shingu (bus timetable 4) (~1540 yen)
[Train] Shingu to Nagoya (Wide View Nanki) (~6870 yen, non-reserved seat)
[Shinkansen] Nagoya to Tokyo

By bus, airplane via Shirahama

Time: ~4 hrs (depends on connection times)

Route Explanation Tokyo to Hongu (Yunomine Onsen, Kawayu Onsen):

This route requires one flight, one taxi, and one bus ride.

[Airplane] Tokyo (Haneda) to Nanki-Shirahama
[Taxi] Nanki-Shirahama to Kii-Tanabe station (~4000 yen)
[Bus] Kii-Tanabe to Hongu area (bus timetable 1) (~2000 yen)
or
[Airplane] Tokyo (Haneda) to Nanki-Shirahama airport
[Bus] Nanki-Shirahama airport to Shirahama station (~340 yen)
[Train] Shirahama to Kii-Tanabe station (~200 yen, local train)
[Bus] Kii-Tanabe to Hongu area (bus timetable 1) (~2000 yen)

Route Explanation Hongu (Yunomine Onsen, Kawayu Onsen) to Tokyo:

This route requires one bus, one taxi, and one flight.

[Bus] Hongu area to Kii-Tanabe (bus timetable 2) (~2000 yen)
[Taxi] Kii-Tanabe station to Nanki-Shirahama (~4000 yen)
[Airplane] Nanki-Shirahama to Tokyo (Haneda)
or
[Bus] Hongu area to Kii-Tanabe (bus timetable 2) (~2000 yen)
[Train] Kii-Tanabe station to Shirahama (~200 yen, local train)
[Bus] Shirahama station to Nanki-Shirahama airport (~340 yen)
[Airplane] Nanki-Shirahama to Tokyo (Haneda)