Here is the full list of all 13 Tobe's Vertical Adventure achievements.
Clear Tobe's Story Mode
Collect All Treasures With Any Character
Clear Nana's Story Mode
Rescue All Animals With Any Character
Nana escapes each of the crumbling Mystic Caves within a total of 115 seconds
Clear Co-Op Story Mode
Tobe escapes each of the crumbling Mystic Caves with all treasures and animals within a total of 120 seconds
Tobe escapes each of the crumbling Frozen Cavern with all treasures and animals within a total of 130 seconds
Tobe escapes each of the crumbling Phoenix Cave with all treasures and animals within a total of 145 seconds
Tobe escapes each of the crumbling Grand Pyramid with all treasures and animals within a total of 190 seconds
Nana escapes each of the crumbling Frozen Cavern within a total of 90 seconds
Nana escapes each of the crumbling Phoenix Cave within a total of 95 seconds
Nana escapes each of the crumbling Grand Pyramid within a total of 145 seconds