На заправке я решила заправиться до полного бака бензина, чтобы потом не волноваться.

На заправке is the usual way to say at the gas station in Russian. Many “places/locations” are idiomatically used with на (on/at), e.g. на вокзале, на почте, на работе, на заправке.
Here заправке is Prepositional case (used after на when it means location: где?).

Yes. решила is past tense feminine singular of решить because the subject is я spoken by a woman (or a feminine narrator).
If a man says it: я решил.
Plural: мы решили.

решить (perfective) = to decide as a completed decision (one-time result). That fits here: she made a decision at that moment.
решать (imperfective) would be used for an ongoing process or repeated decisions, e.g. я долго решала, что делать (I was deciding for a long time).

заправиться is the common everyday verb meaning to get fuel / to fill up (one’s car)—literally “to fuel oneself,” i.e. your vehicle.
заправить is transitive: заправить машину (to fuel the car), заправить рубашку (to tuck in a shirt), etc.
So заправиться focuses on the action you do for yourself: “I filled up.”

заправиться is perfective because it’s a single completed action (the goal is to finish fueling).
заправляться (imperfective) would be used for:

• a habitual action: я часто заправляюсь тут (I often fill up here)
• an action in progress: я заправлялась, когда он позвонил (I was filling up when he called)

до + Genitive means up to / until (reaching a limit).
So до полного бака = until a full tank (i.e. “to the point where the tank is full”).
That’s why полного and бака are Genitive.

Because до requires Genitive: до чего? → до полного бака.
Inside that phrase, бензина is also Genitive because it’s a “quantity/container + substance” pattern:

• бак чего? → бак бензина (a tank of gasoline)
  So the whole chunk is: до (полного) бака (бензина), all Genitive as needed.

Both are natural.

• до полного бака is very common and usually understood from context (at a gas station it’s obviously fuel).
• до полного бака бензина is just more explicit; it can sound slightly more “complete/technical,” but still normal.

чтобы introduces a purpose clause: in order to / so that.
After чтобы, Russian often uses an infinitive when the subject is the same:
я решила …, чтобы потом не волноваться = “I decided … so I wouldn’t worry later.”
не волноваться (imperfective) describes the general state/process of worrying, not a single “completed worry.”

With verbs, не is written separately: не волноваться, не делать, не знать.
(There are rare exceptions, but for normal verb negation it’s separate.)

потом = later / afterwards.
Its position is flexible. These are all possible with slight emphasis differences:

• …, чтобы потом не волноваться (later = after fueling)
• …, чтобы не волноваться потом (slightly more emphasis on “not worry”)

Word order is fairly flexible because endings show roles.
Both are correct:

• На заправке я решила… (sets the scene first: “At the station…”)
• Я решила на заправке… (starts with “I decided…”; location is secondary)
  The original is a very natural “context first” order.