Strafe in Höhe von 30 € - Penalty in the amount of 30 €; makes more sense concerning the context.

bekommen bekommt would be the presence form. Since this is perfect tense, "Bekommen" is the right conjugation.

der - "die" is the female article, but since you're male you'll have to use "der" . ;]

es -"diese Situation" isn't fitting so well in this context, it just sounds unnatural. Instead it's more suitable to simply use the pronoun "es", which will stand for the mentioned incident.

Nun werde ich erzählen, wie es dazu kam. You usally don't say "Ich schreibe um eine Erklärung abzugeben", so I altered things to make it a bit more natural : "Now I will explain how things / how it happened".

bester  Again you used the female suffix instead of the male one.

wohnt seit sechs Monaten in München   - Word order Error

versprach, hatte,

ihn im Sommer besuchen würde - You used "in den" Sommer, but here you would have to use "im", which is the the contraction of "in dem". You also made a grammatical mistake, this sentence calls for Plusquarmperfekt as the action of your promise happened before you went to Munich. Thus, "versprach" should actually be "versprochen", "habe" has to become "hatte."

gelangte - "fuhr" is the past form of "fahren". The word "fahren" is basically a merge of the English verbs "to drive" and "to go by", but there's one thing to pay attention to. When you're going by plane, "to fly" / "fliegen" is used instead. Since you already used "flight" ("Flug") in your sentence I used the preterite of "gelangen" instead, which means "to reach / to arrive somewhere".

erkannte - In terms of grammatic it's fine, but concerning this context it sounds rather overblown and unnatural, "erkannte" would be the better choice here.

dem - "Auf dem Rückweg" is dative, that's why it has to be

beschloss ich   - Word Order Error

dass ich wäre, es nehmen   - This phrase is not grammatically correct, to be fair I'm not sure what exactly you're trying to say here. Feedback would be nice.

den   - Dative again, "den" instead of "der", "der" instead of "die".

der 

um -  unnecessary here.

vergaß   - You usually spell it like "vergaß", not sure if you're capable of writing it that way on your keyboard.

vergaß ich - Word order Error

einer - Dative, "einer" instead of "eine".

neuen  - Dative, "neuen" instead of "neue".

ohne die Zeit zu wissen  - I think you were trying to say something like "without knowing the time" (?) so I translated it that way. The way you wrote it just meant "without time".

Good luck on your essay, let me know if I could help you out.

Ich habe neulich eine Strafe in Höhe von 30 € bekommen. Ich bin ein dreiundzwanzigjähriger Student aus England, der im Urlaub war als es passierte. Nun werde ich erzählen, wie es dazu kam.

Mein bester Freund wohnt seit sechs Monaten in München und ich hatte ihm versprochen, dass ich ihn im Sommer besuchen würde.

Ich gelangte mit dem Flug nach München und erkannte nicht, wie teuer ein Taxi zum Zentrum war. Auf dem Rückweg beschloss ich [...]. In der Eile den Flughafen zu erreichen, vergaß ich eine Fahrkarte zu kaufen. Ich war in einer neuen Stadt, ohne Landkarte und ohne die Zeit zu wissen!