I understand Austria was excluded in Bismark's Germany, not allowed to join Germany after WWI due to the Treaty of Saint-Germain-en-Laye, included as part of Hitler's Anschluss, and then split again after WWII.
According to Wikipedia's article on the Germany Reunification:
Article 146 was amended so that Article 23 of the current constitution could be used for reunification. After the five "New Länder" of East Germany had joined, the constitution was amended again to indicate that all parts of Germany are now unified. Article 23 was rewritten and it can still be understood as an invitation to others (e.g. Austria) to join, although the main idea of the change was to calm fears in (for example) Poland, that Germany would later try to rejoin with former eastern territories of Germany that were now Polish or parts of other countries in the East.
Was there any serious movement for Austria to join with the newly reunited German state, either before 1990 or before?
Not really. While there always was some pro-German sentiment(particulary among right-wing parties), there never was a mainstream movement. This is mainly due to the fact that such a unification is expressly prohibited by the 1955 national treaty signed with the Allied nations.
Source: [National Treaty] (https://treaties.un.org/doc/Publication/UNTS/Volume%20217/v217.pdf)
The relevant article is on page 235, article 4.
edit: Autria applied for membership in the EU in 1989(and was admitted in 1995), there already was some kind of unification.
No. For several reasons.
First, because technically a "national or economical union" is still prohibited in the Austrian State Treaty (which the USSR used for a long time to keep the country out of the EU) and thus the Austrian constitution. Granted, there have been some parts that were voided, specifically the articles 12-16, covering such important topics as banning former Nazi party members from joining the Austrian Army (art.12) and prohibiting the country from training German pilots (art.15) or purchasing German or Japanese aeroplanes (art.16), but only after the dissolution of the USSR and in theory the signatory parties to the treaty (France, the UK, the US and, as successor state to the USSR, Russia) still hold veto powers on changes to the State Treaty.
And second, because at that point the Austrian identity had once again been cemented in all but possibly the very oldest generation. After the first World War the tiny rump of the former Empire would gladly have joined Germany because they could not imagine being able to survive on their own. At the time of the Anschluß in 1938 many Austrians still regarded it very positively (not the official 99% yes-votes but considerably none the less). However, with the atrocities of the Nazi regime the Austrian population was very eager to distance itself from Germany as much as possible and while anti-German sentiment returned to normal over the next decades the Austrians had already established themselves as distinctly not just-Germans with a good international standing based around its neutrality and position right at the border between the First and Second World, which led to things like Vienna hosting several meetings between US and USSR leaders and becoming a permanent seat of the UN.
By the time the 90s came around only a tiny fringe that probably largely corresponds with local neonazi groups would have been in favour of joining Germany.