I have full names from the game (simply watch the playthrough and match correct names instead of track number).
Unfortunately, almost all songs are present in other levels...
01 - Title Screen
02 - Character Select
03 - Level Complete
04 - Street Chase & Hospital & Hang-Glider
05 - To the Cabin
06 - Cabin & Secret Caves & Castle Osaka
07 - Nurse Shubuya & Sumo & Cave of Gold - Koga
08 - Bonus Round & Konang Dojo
09 - Woods & Japanese Garden (1)
10 - Hillside & Japanese Garden (2)
11 - Credits (the weirdest credits I've ever seen).