Hi Tandel,
Per my understanding, you might want to display values of other columns of ListA based on the selected option of the Lookup column in the new form of ListB.
As there is no OOTB feature can meet this requirement, I suggest you create a custom solution by applying some custom HTML and JavaScript to the new form(NewForm.aspx).
With JavaScript, we can monitor the change event of the drop down menu and get the users selection, then query the ListA with the selection with the help of SharePoint
JavaScript Client Object Model for the values of other columns of a specific item.
About how to get user selection of a Lookup field using JavaScript in new form:
https://social.msdn.microsoft.com/Forums/office/en-US/529ee28f-80bf-44a6-9cb4-392e74213076/how-to-get-value-from-lookup-field-on-onchange-event?forum=sharepointdevelopmentprevious
How to: Create, Update, and Delete List Items Using JavaScript
http://msdn.microsoft.com/en-us/library/office/hh185011(v=office.14).aspx
Common Programming Tasks in the JavaScript Object Model
http://msdn.microsoft.com/en-us/library/office/hh185015(v=office.14).aspx
With the needed values retrieved, you can then generate custom HTML tag to display in the new form.
About how to generate HTML with JavaScript:
http://blog.schettino72.net/posts/javascript-html-generation.html
Adding JavaScript and CSS to SharePoint
http://techtrainingnotes.blogspot.com/2012/05/adding-javascript-and-css-to-sharepoint.html
Thanks