Yes, you can send an alarm to someone else's phone. There are a few different ways to do this, depending on what type of phone the other person has and what type of alarm you want to set up.
If both you and the other person have iPhones, then you can use the built-in Reminders app to set up an alarm. Open the Reminders app and tap on the + button in the top-right corner. Enter in the task or event that you want to be reminded about, then tap on the i icon next to it. On the next screen, scroll down and tap on 'Alarm'. From here, you can choose when you want the alarm to go off, and whether you want it to repeat. Once you've got everything set up how you want it, just tap on 'Done' in the top-right corner.
If you have an Android phone but the other person has an iPhone, there are a few different apps that will let you send alarms between phones. One option is Alarmy (Sleep If U Can), which is a free app that lets you set up alarms that go off at specific locations or times. Another option is IFTTT (if this then that), which lets you create 'recipes' for all sorts of things - including setting off an alarm on another person's phone.
Of course, if neither of you have smartphones then there are still options available - like setting up a regular old fashioned alarm clock!