This is the classic dilemma when substituting any simple switch for a powered/sophisticated switch. Many light switches in legacy (pre-NEC 2011) homes are wired with switch loops, which bring only "always-hot" and "switched-hot" to the switch - no neutral since plain switches don't need them.
You have what is very common - that switch loop with always-hot and switched-hot. The installers did what is now a Code violation by failing to mark the white wire as a hot on both ends. Code also requires the white be used for always-hot if it's not being used for neutral (so it's easier to detect).
The upshot is that you cannot use a "smart switch" unless that smart-switch is designed to not require a neutral wire. And this is something you better get used to, as it's probably true for half the light circuits in your house. As it happens, I just yesterday saw a listing for a GE pushbutton timer of the type you seem to want, which did not require a neutral wire but did require a ground wire. That might suit your needs.