Can you pay life as part of a cost if you only have 1 life?

You are allowed to pay life as part of a cost even if you only have 1 life remaining. CR 118.4 states that if a cost requires paying life, you simply pay that life — there is no rule preventing you from paying life when doing so would bring you to 0 or below.

However, state-based actions are checked immediately after a player receives priority again following the resolution of the ability or spell. CR 704.3 and 704.5a specify that a player with 0 or less life loses the game the next time state-based actions are checked. So while the cost is paid legally, you will lose the game very shortly afterward unless something prevents it (e.g., an effect that says you can't lose the game).

It is important to note that the cost is still considered paid in full. The spell or ability continues to resolve normally — your impending loss of the game does not counter or cancel it. Your opponent cannot respond before state-based actions are checked after the spell/ability resolves.

Concrete example: You have 1 life and activate Necropotence's draw ability by paying 1 life (in a hypothetical variant where it has an activation cost). The cost is paid legally, the ability goes on the stack. After it resolves, state-based actions are checked and you lose the game at 0 life — but the ability fully resolved first.

The key rules here are CR 118.4 (paying life as a cost), CR 704.3 (when state-based actions are checked), and CR 704.5a (a player with 0 or less life loses the game).