How do we know what our solar system looks like?
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You have confused the Solar System with the Milky Way Galaxy. The Solar System is the Sun and the planets and asteroids, comets, etc. It lies within the Galaxy. The Galaxy is many many times larger and contains billions of stars like our sun. The spiral arms are typical of our type of Galaxy. Although we cannot see it (from outside as you have stated) we can see similar galaxies to ours in the night sky for example the Andromeda Galaxy which is like a sister galaxy to the Milky Way and approx 2 million light years away, yet you can see it with the naked eye in the night sky.
Telescopes and satellites.
You seem to be confusing the terms 'solar system' and 'galaxy' here. Assuming you mean the galaxy, we can see an awful lot of it from where we are, and we can compare that with other galaxies we can see to derive a likely overall structure.
It sounds like you're asking about our galaxy and not the solar system. We look at samples of stars (large samples) and determine their distances. Then if we plot their locations, we get a good idea of what it would look like.
Our Solar system or our Galaxy? You're talking about spiral arms so I'm thinking your talking about our Galaxy and not our Solar system. Well don't know and we never will know what our galaxy looks like from afar, but from observations of other glaxies we know there are basically three type, Spiral Galaxies, Like andromeda and the milky way, Eliptical Galaxies like M32 and M110 (these are usually "dwarf" galaxies, and irrigular Galaxies.
When you look into the night sky from a dark place on a clear night you'll see what appears to be a band running accross the sky from roughly north to south. That is our galatic plane, we are looking at the disk "edge on" and we know from observations from other glalaxies that Spiral are the one of the three that have this thin disk. So we can extrapolate that since we are seeing a disk structure of our galaxy then that it must be a spiral.
Solor systems dont have spiral arms.
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