When reaching out to former colleagues and managers for LinkedIn reviews or grad school recommendations, it is best to be direct and straightforward in your communication.
There is no need to pretend to catch up before making the request, as work relationships operate under different etiquette than social relationships.
It is normal to contact people out of the blue when job-related matters arise, even after long periods of no contact.
When networking, remember that there is usually an assumption of give-and-take, and maintaining relationships can lead to mutual benefits.