Yes the age of a home does affect your homeowners insurance rates and what you'll pay per month for your homeowners insurance policy.
Older homes are usually much more expensive to insure than newer homes are.
The reason that older homes are usually higher in cost to insure is because it will cost more money to replace your older home with a newer home should your home be completely destroyed beyond repair.
So when you have an older home you can sometimes expect to pay a much higher monthly cost to insure that older home.
Newer homes will usually be much cheaper to insure than older homes are because they are less likely to need replaced or need many repairs on them.
Your neighborhood and your credit score also factors into the cost of your monthly insurance rates as well.
If you bundle your homeowners insurance with your auto insurance or other insurance you can usually get your homeowners insurance cheaper.