I am in a similar situation, but in the process of weaning the almost-4 year old off of expecting screen time every time little sister (almost 2) takes a nap.
From when she stopped napping around 2 1/2, I would always put a video on for her while I got the baby to sleep. If I knew I was going to sleep too, I would make it a longer video. This has become an expectation and an entitlement. I also realized that older DD still needs occasional naps, though she would much rather watch a video.
I started out using different things to entice older DD to lie down, when she was 3 1/2. I got her cute kiddie earphones, and let her listen to a CD while she laid down. She was allowed to lie on the couch with her blanket, so it felt like a fun thing to do. The first few times I did this she actually fell asleep. But then she got wise- she didn't want to do it anymore, because she really didn't want to sleep. She was back to demanding videos. And I had to convince her, I couldn't just say "no" and leave it at that, because she would come into the room where I was trying to get little sister to nap (and she's a light sleeper). Actually this was only really an issue with me on the weekends- I was working full time by this time, so we had an au pair taking care of the kids. And for some reason older DD was much more willing to sit around and read books in the living room while the au pair put the baby to sleep (which sometimes took a long time). But I got the au pair in on trying to get DD to nap when she needed it.
The next thing we did was use something fun happening in the late afternoon as an enticement for rest time. I would tell DD that she had ballet class, but she needed to lie down for 45 minutes if she wanted to go. I would make up stuff we were going to do- playground outings, baking cookies, visiting a friend, going out to dinner, ANYTHING that I thought would work.
And now we're at the point where I can basically insist that she lie down, no video, and she will fuss and maybe throw a mini-fit at first but as long as I calmly hold to my guns she will do it. Usually. It is a problem when my husband is around, because he has a harder time sticking wtih no and she knows it, so it has become a vicious cycle where she will not take no from him because she knows he eventually gives in. So he usually puts a video on for her when I'm napping wtih the baby, unless he has time to engage her in something else. One time I napped with the baby and woke to find that DH took older DD out with him grocery shopping. Awesome :).
We have a new au pair now, and I neglected to give her the password for my laptop the first week. But I saw that she let DD use her netbook to watch a video when she was with younger DD around naptime. Bah. We'll have to have a talk about that.
The funny thing is, both of my kids are great at entertaining themselves usually, it's just when they have the expectation of getting to watch a video they suddenly forget how to play with their toys. I think that's the insidious thing about screen time.