I believe that it depends on the day of the week that you plan to visit and the time of your visit. We made a reservation for the opening time (I believe it was 9 am) and arrived 30 mins prior as directed on the reservation. There were 10 people there in addition to a few large tour groups by the opening time so I'm not sure if we needed the reservation or not. The large groups did the shorter tour and we opted for the longer tour. It was really nice having the apartments to ourselves without a crowd. We stayed near the opera and I think it was a 15 or 20 minute subway ride and a short walk to get there.

Make sure to allow time to wander in the gardens. They were lovely in May and I bet they are nice in April as well.

We were there mid-week in early May and went first thing - we had reserved tickets but at that hour and time of week it really wasn't necessary. It took about 20 to 30 minutes from the inner ring, using the U-bahn and a short walk, if memory serves.

I don't think advance tickets would be necessary. We went in early June without advance tickets and didn't have a long wait for a guided tour.
I was at Schonbrunn in early March (to walk the grounds in the snow) and saw that the building had a significant amount of scaffolding and they were also doing a lot of construction in the front. I don't think it will have an effect on the interior tours, but the exterior isn't looking at its best.

When I was in Vienna back in July/2000, there was no need for reservations. I went there on a Sunday morning and had no problem with long lines to buy tickets and to get in.

I have gone through the palace twice and neither time did I need advance tickets.
I do suggest that you take the 40 room tour, if that is the one with the maximum number of rooms. We also used the audio guide, which was of course in just about any language most of the world would need.

There are also guides around with badges listing the languages in which the wearer is fluent. I saw some interesting combinations: Russian, Hungarian, Turkish, and German for example. Of course there were a good supply of well-informed guides wearing badges that listed English among other lanugages.

Walking the gounds is interested. And a walk up the hill to the Gloriette is worth if for no other reason than to take in the view.

It is a little steep, and I can see why Maria Theresia had to be hauled up in her declining years. But I made it at age 67 with no trouble.

The greenhouses are worth the visit, as well.

The U-Bahn takes you to the Schönbrunn Palace stop. The stop is very convenient to reach from down town Vienna.