If you take the first cable car up to Klein Matterhorn and you are already well acclimatized and you're fit, then you can do all that in a single day. Otherwise, there's plenty of options for accommodation. There's the Refuge Des Guides D'Ayas or the Klein Matterhorn Lodge.
If you want a head start you could try to go for the Rossi and Volante Bivouac (the one P. Vis was hinting at). Just know that it's often very very crowded. Up to the point that you might not have a spot inside.
Depending on whether you'd like to climb the both summits of the Lyskam, it'll be easier or harder. Climbing Castor and after that Felikhorn will be really easy if you have some experience with this kind of climbing and are reasonably fit. If you start early and make good progress you'll have time left to continue. Otherwise you might want to descend a bit again and sleep at the Rif. Quintino Sella.
Lyskamm might be slightly more difficult, but not much more. The ridge is very exposed, so go slow and safe.