I need to execute that transaction alongside my own transaction (kinda wrap two transactions into another) so if either of them fail, none should get executed.
This should be possible. Technically, you would be executing two instructions inside a single transaction, and the transaction would revert if either of the instructions reverted. This distinction is shown in the below diagram, taken from here:
Simply, instructions are what actually make state changes in the blockchain, and can be thought of as calls to programs. For example, a user can transfer their SPL tokens through a transfer instruction. Transactions are packages of instructions that users send to validators.
A tutorial for sending multiple instructions in a single transaction can be found in the Solana Cookbook.