While this is an interesting question the answer is ‘No.’
You cannot apply for an Employment Visa for yourself. Employment and residency in UAE has to be sponsored by your employer. An employer is the appropriate party to apply for an employment and residency visa for employees. This means you need to be hired prior to the application for your employment visa. Which means that you need to pass the interview and get an indicative employment contract.
By indicative employment contract what I mean is that, all the companies coming under MOL(Ministry of Labour) will have to provide an offer letter that is generated by the MOL. This is usually called as Labour Contract. As an employee, it is recommended that you read the contents of the agreement thoroughly because once it is signed, there is no way that the terms and conditions can be changed or amended without both parties agreeing to it.
While this applies to all companies in the purview of the MOL, Freezones such as DIC, DMC etc also have started similar process. Freezone visas are directly sponsored by the respective authorities.
Semi Government companies have a different set of rules and regulations on this.
Click here for a List of Companies in UAE.