発 is a counter for shots/launches (gun, rocket, arrow, fireworks, etc), and, by extension, trials/attempts. For example 一発で成功する means "to succeed on the first attempt", and 一発合格 means "pass on the first try". Likewise, 一発解決 describes how your trouble will be solved quickly without trying many options. So "instant" is close to this 一発, but there may be a better catchy word.
As for the "missing" particle between 一発解決 and お悩み相談, they are treating お悩み相談 as the "main" title and 一発解決 as its catchphrase or "subtitle". Just like
title: subtitle is a common format in English (e.g, "Mad Max: Fury Road", "2001: A Space Odyssey"), in Japanese, attaching a subtitle/catchphrase-like word directly before a name is a common way to form an interesting title (e.g., 名探偵コナン, 燃えよドラゴン, ジャングルの王者ターちゃん, わくわく動物ランド).