Because \labday uses \addchap (KOMA-Script's unnumbered version of \chapter), you have to patch this command. I added a \par at the start of the changed definition in case (as in your example) no \par or blank line is added before \labday in the document body.
\documentclass[oneside]{labbook}
\usepackage{etoolbox}
\makeatletter
\patchcmd{\addchap}{\if@openright\cleardoublepage\else\clearpage\fi}{\par}{}{}
\makeatother
\begin{document}
\labday{Monday}
First day in the lab.
\labday{Tuesday}
Second day in the lab.
\end{document}
Want you want, can be achieved with one of usual packages for headers and footers. I decided for this one, which is shipped out with KOMA-Script and does, of course, integrate best into the KOMA-Script class scrbook. It is scrlayer-scrpage (could be by the way also used with other classes not part of KOMA-Script).
NOTE: My example does only work from the recently published version 3.12 of KOMA-Script on. Perhaps you must update your TeX distribution first.
\documentclass{scrbook}
\usepackage[markcase=upper]{scrlayer-scrpage}
\pagestyle{scrheadings}
\clearpairofpagestyles
\ohead[]{\pagemark}
\cfoot[\pagemark]{}
\chead[]{\headmark}
\setkomafont{pageheadfoot}{\scriptsize}% or "\footnotesize" depending on fonts
\automark[chapter]{chapter}
\automark*[section]{}
\usepackage[english]{babel}% added for "blindtext", but useful anyway
\usepackage{blindtext}
\begin{document}
\title{A Title}
\author{Joe Miller}
\maketitle
\chapter{From Aa to Zz}
\Blindtext[15]
\Blinddocument
\end{document}
For description of commands and possibilities of more fine tuning see documentation of scrlayer-scrpage (chapter 6 in recent KOMA-Script manual). The command
\chead[]{\headmark}
acts as if I had written
\cehead[]{\leftmark}
\cohead[]{\rightmark}
I also intentionally added a chapter without a section.
You should know, that the use of uppercase letters in headings is typographically questionable. I strongly recommend you to read, what Markus Kohm wrote regarding this in the documentation in the section of the package option markcase. (For reasons of AMS decision read Barbara Beeton’s comment below, and the addition.)
For small caps instead of full uppercase letters (typographically better, but kerning issues are in most fonts still visible for experienced readers) use this (only changed lines):
\usepackage{scrlayer-scrpage}
\setkomafont{pageheadfoot}{\scriptsize\scshape}
The class option numbers=enddot lets appear dots after numbers.
When one uses the class option chapterprefix=true (on/yes), but do not want the prefix in headers, and also if one wants a dot after numbers in headers, but not in titlings, the command \chaptermarkformat must be redefined:
% Remove the chapter prefix:
\renewcommand*{\chaptermarkformat}{\thechapter\autodot\enskip}
% Enforce additionally a dot after numbers in headers:
\renewcommand*{\chaptermarkformat}{\thechapter.\enskip}
Best Answer
AFAIK KOMA-Script does not provide something like
\chapterstyle. Nevertheless there are some options that influence the output and behaviour of chapters, see Section "3.16 Structuring of Documents", e.g.:open={any|right|left}chapterprefix={true|false},appendixprefix={true|false}headings={big|normal|small|...}