That sentence does not work. You have to say "jag har inte en katt". Commonly, "inte" goes after the verb it modifies, BUT in a subclause it goes before the verb.
Heres an example, with main clause and subclause separated:
"Han säger inte | att han inte kan komma"
"He's not saying | that he can't come"
The purpose of this sentence is to learn 'ingen', one of the three forms of 'no' (=none). "Han har ingen katt"="He has no cat" The other translations do have the same meaning but are translations of "Han har inte en katt". Too bad, because then you have not learned the difference between "ingen/inget/inga" and "inte".
"He hasn't a cat" is archaic but perfectly acceptable. "He hasn't got a cat" is more colloquial and rather redundant.
"He hasn't got a cat." "He doesn't have a cat." "He hasn't got any cats." "He doesn't have any cats." These versions should all be accepted.
"any cat" is a grammatically wrong construction. After ANY, all countable nouns have to be in the plural. Same applies to "no cat".
"He hasn't a cat." is not standard grammar so it should be avoided in a language course.
inte (= not) is used to negate verbs: Han äter inte 'He is not eating'.
ingen/inget/inga (= no) is used for negating nouns. Which form depends on the gender and number of the thing you don't have.
en gender words - ingen
en katt - Han har ingen katt ('He has no cat')
ett gender words - inget
ett hus - Han har inget hus ('He has no house')
plural for both genders - inga
Han har inga katter ('He has no cats')
Han har inga hus ('He has no houses')
Is there an easy way of telling the gender of a word/noun? For example a general rule in Spanish is that if the word ends with or has the main syllable contain an a it is feminine.. for eg "la mesa" (the table) "la puerta" (the door) whereas masculine words generally end in or have an o in them as the major syllable "el techo" (the roof) el sillón (the couch/lounge).. (of course there is always exceptions in every language). Is there a general phonic rule like this for Swedish please?
I wrote He hasn't any cat - he (han) has(har) not any (ingen) katt (cat) and it was wrong. Read most of the comments but still don't understand why "he hasn't any cat" does not work? How would you say He hasn't any time or money for example? Or must you always use ingen (inget, inga) as "no" instead of not any?