There are several ways to update your PSP, please check below:
Network Update: You can use the PSP’s Network Update feature to update your PSP if you have access to a Wi-Fi hotspot (wireless internet connection). The update feature will allow you to download the latest firmware version and update your console.
Update via PC: If you do not have a wireless internet connection, you can download the update onto your PC and transfer the update file onto your PSP. The updates can be downloaded from Your PSP.
UMD Update: Some UMD games come with a firmware update installed on the disk. You can update your PSP by accessing the UPDATE feature under the GAME menu on your PSP. Firmware versions vary on the disk depending on how old the game is.