You can see if there are any startup programs you don't want or need and disable those (don't disable intel, amd, antivirus, microsoft, or nvidia programs). Additionally, you can buy a solid state drive. For a hard drive, a minute actually isn't all that bad of a startup time, especially for a laptop which usually has a slower hard drive to save power.
You can also put the computer to sleep instead throughout the day and only shut it down when you're done with it for the day, but it will consume battery while in sleep mode (just at a greatly reduced rate).