Maybe, unless there is a problem with your iPhone's backup and it sounds like there is a problem with it or you wouldn't have lost any contacts.
Although contacts are included with your iPhone's backup, contacts are designed to be synced with a supported address book application on your computer and should be available on your computer with or without an iPhone or any cell phone. Not a good idea to depend on your cell phone alone for this data, which can be lost or stolen.
To restore your iPhone with iTunes from your iPhone's backup, select Restore under the Summary tab for your iPhone sync preferences with iTunes. When prompted to update your iPhone's backup, as the first step after selecting Restore, decline this prompt.
Go to iTunes > Preferences. Under the Devices tab, how many backups are shown? If more than one, what is the date and time shown for each backup? If there is more than one backup, you may not want to select the current backup - select a previous backup when restoring your iPhone with iTunes. After the firmware is re-installed, you will be prompted to restore from the backup, which is the default selection. If there is more than one backup available, the most recent backup will be selected by default. If more than one backup is available, you can select from another or previous backup.