It is not possible to restore a snapshot, which will create a new instance. It is not possible to restore a snapshot without creating a new instance.
To avoid loosing connection with your web page, what many developers do is to have both instances (the original one and the newly restored one) running at the same time. One of them (the original) will have the domain pointing to it, the other one will not be used yet. This way, you only need to change the DNS to point to the new instance.
Note that by doing this, you should have your database stored in another instance or engine, like RDS.
Let us know if you have any question about this.