We emailed the Hungarian embassy in Russia. They told us they accept visa applications through VFS global and when we checked, they had appointments in a week, so they look to be the best bet. They also seem to issue visas the quickest (2-3 weeks). For first-time Schengen visa applicants, we were advised to request 15 days, as it's considered a realistic vacation timeframe and improves the chances of approval. However, if your wife has had multiple Schengen visas, you might be in luck to just apply now and you'll get a 6 month visa like Pirozhok did.
Our plan was -- wait until the interview letter, then immediately "plan our Hungarian vacation" and schedule the visa appointment in Moscow, and also schedule the medical examination for the day before the visa appointment (so that we could finish that and then just surrender my husband's passport to the Hungarian embassy without worrying about getting the medical exam in time as well).
@SpaceCat From what I understand, they only let you reschedule after you miss the interview, that's all I've seen anecdotally.
However, we have had a change in our plans because NVC put us in Yerevan, Armenia instead of Warsaw... and I thought it was going to update after DQ, but they've kept us in Yerevan.
I wish you luck, and let us know how it goes.