Yes - it's the thing I notice the most often with non-native English speakers (who still have brilliant English) - they will get prepositions wrong because it's really just a random pick that is very hard to pick up unless you have been speaking that language nearly constantly for at least a year or two. For example "On Thursday, in April, at the park." Non-native speakers will often say "At Thursday, in April, on the park." or something.
Helpful hint for åt, för, and till (the three prepositions which translate to "for" in English) ...
åt is used if it's being done as a favor (as in this sentence) för is used if it's done before an audience (han sjunger for oss ... he sings for us) till is used if it's being done as a gift (han köper ett bord till henne ... he buys a table for her)
I think the first a in bara is more like "ah," while the ä in bära seems to often be more like the a in baa, although not so broad. At either times (according to the Duo pronunciation feature) ä sounds to me almost like "eh." I think, as in English, it depends whether or not the syllable containing the vowel is stressed or not, or in a word that contains stress, but I'm not sure. Sometimes (although not always) it's helpful to type similar sounding words into Google translate and then listen to them being consecutively pronounced.
ä normally = eh (ett äpple = ett epp-le), but it doesn't seem to be there are exceptions. e.g. "Jag är" = sounds basically like "Yarg ahhr" and not "Yarg air". bära and bara seem to be basically the same, though I haven't heard for a while.
Also, you'll notice I asked for a native speaker to explain it to me... Du svarade men det inte samma sak. ;-)
Added that, but note that case in many cases can refer to different types of bags/things, like en portfölj 'a briefcase' or en verktygslåda 'a toolkit' or ett fordral (the case for e.g. a cell phone). So as a general translation, it's obviously a less good choice. The farther you go from the most common words, the bigger the risk that we haven't yet added a word as an accepted answer, even if we want to include it. We add words all the time so keep reporting (using the Report a problem button) but remember that whenever you use a rarer word, chances are your answer won't be accepted.
While I'm at it, the first translation for 'case' as an object would be låda, for instance en låda vin is 'a case of wine'.
I presume its contextual, particularly since they're different part of a sentence. If åt comes afterna verb it means 'for'. If it is in the place you'd expect a past tense verb (e.g. the second position in a past-tense sentence), it will be 'ate'.
Take the sentence "jag åt åt honom". That could only mean "I ate for him", and can't mean "I for ate him".
Similarly, "jag åt med henne" can only mean 'I ate with her' (because åt is in the verb place), and "jag sprang åt henne" can only mean 'I ran for her'.