To process one table, try something like this:
create procedure MyProcedure
@table_name varchar(100)
as
begin
declare @s varchar(max) =
concat('select * into ', quotename(@table_name + '_Old_' + convert(varchar(max), getdate(), 120)),
' from ', @table_name, ' truncate table ', @table_name)
exec (@s)
end
Such things will probably not work in case of relationships between tables.