Don't import your data directly into tblCustomer. Import it into a temp table that has the same structure as tblCustomer. Make sure that in tblCustomer that Customer ID is a unique primary key.
After you import the data into a temp table, then use an append query to append the new data from the temp to tblCustomer. Because the Customer ID is a primary key set to INDEXED - NO DUPLICATES, Access will reject the duplicates based on Customer ID.
If your spreadsheet doesn't have Customer ID, then you'll have to create a unique key based on a combination of Last Name + First Name + Address or something like that.
Michigan Dept. of Environmental Quality, Air Quality Division