(I rewrote the answer more or less completely after the OP provided an important piece of information, viz., that the bibliographic items of his/her document must be formatted according to APA guidelines.)
If your bibliography has to be formatted according to current APA guidelines, you should be using the apacite bibliography style, along with the apacite citation management package. FWIW, the apalike bibliography style is ca 30 years old and, as such, does not implement current APA formatting guidelines...
The good news is that the apacite style -- unlike the apalike style -- knows what to do with fields named day and month and recognizes an entry type called @newspaper. All you really need to do, mainly, is to change the entry type from @article to @newspaper. If you want to use natbib-like citation commands, you should load the apacite package with the option natbibapa.
Incidentally, if you change the argument of the month field from {July} to jul (note: no curly braces), you'll get the language-appropriate form of the month and day fields ("July 19", "19. Juli", "19 juillet", etc) automatically if the babel package is loaded with a suitable language option. Thus, do make a habit of using only jan, feb, mar, etc as the arguments of the month field.

\RequirePackage{filecontents}
\begin{filecontents}{sqwg2.bib}
@newspaper{Jurgensen,
author = {J{\"u}rgensen, Nadine},
title = {{Die Geb{\"u}hren geh{\"o}ren abgeschafft}},
journal = {Neue Z{\"u}rcher Zeitung},
year = {2012},
month = jul,
day = {19},
pages = {11},
}
\end{filecontents}
\documentclass[12pt]{article}
\usepackage{times}
\usepackage[natbibapa]{apacite}
\bibliographystyle{apacite}
\begin{document}
\citep{Jurgensen}
\bibliography{sqwg2}
\end{document}
apalikebecause I thought this will ensure that all citazions are in apa-style and, on top of that, citations are not recognized if I remove it. I chose@articlebecause of this. I like your solution, but I would prefer to really have the date as one, i.e. (2012, July 19) and right behind the author's name. Is that possible? – Malganas Dec 23 '16 at 09:10yearfield to{2012, July 19}and thenotefield top.~19? (Be sure to re-run BibTeX and LaTeX after changing a bib entry.) – Mico Dec 23 '16 at 13:41\citepcommands shows "(Jürgensen, y 19)" and not "(Jürgensen, 2012)". – Malganas Dec 23 '16 at 16:31\RequirePackage{filecontents}and\begin{filecontents}{sqwg2.bib}to be put in the .bib file or my actual latex? If I try to copy it into my latex file, it causes errors. – Malganas Dec 23 '16 at 16:36filecontentspackage is needed to make my answer code self-contained. You should be editing your real bib file separately, i.e., not as a "prefix" to the main tex file. – Mico Dec 23 '16 at 21:16