A being with 1 lightyear arms or even 1 lightyear across and density similar to flesh would be sufficiently dense to create an enormous black hole. Even planet sized, a being of that size runs into gravity problems, where, lifting it's arm would require significant energy to resist it's on gravitational attraction on it's limbs, so too big doesn't work. But say, really really big. Improbably big, like a two thousand foot tall elephant like animal (low gravity planet, thick atmosphere - who knows). Would it experience time more slowly, or even on an earthly scale, do Elephants experience time more slowly than ants.
Maybe. Muscular reflexes rely on sending a message from the point of contact to the brain and back to the point of contact, so there is a longer transfer time with a larger body. Speed of messages sent by nerves depends on the study and there's a fair bit of variability there (source: http://hypertextbook.com/facts/2002/DavidParizh.shtml) but if we use a speed of 300 feet per second, two thousand feet tall being who steps on a 15 foot tack wouldn't feel it for 6 seconds and wouldn't be able to move for another 6 seconds - so it's entirely possible that a large being would experience the world around it more slowly, but it's also possible that it could develop faster nerve fibers or more localized response systems.
How we experience time is probably more a factor of evolution than anything else. Quick reaction times and faster observation and recognition can improve the odds of survival, so, I don't think there's any guarantee that a large animal experiences time more slowly, but there might be a general truth to that theory. It has more to do with physiology and evolution than physics.
Another possibility is that our reaction time is related to gravitation and the rate of falling. In very low gravity, where you fall more slowly, reaction time to catch yourself, or adjust your body for the fall might be slower and in high G, reaction times might need to be faster, especially since in high G, beings would probably be shorter. My musings on this, are, of-course, purely theoretical but not really science.
It's a reasonable enough question though a being 1 light year across is hard to conceive, but I don't think there's a clear answer. By watching kittens, that they seem to be able to respond faster and do more in a second than larger animals are, so it seems at least, somewhat true that smaller animals experience a second to be longer than larger animals, but to what extent it's true in a grand sense - I'm not sure.
on the last part, let me quote Carl Sagan:
There are naive questions, tedious questions, ill-phrased questions,
questions put after inadequate self-criticism. But every question is a
cry to understand the world. There is no such thing as a dumb question
and this, from dear abby (of all places)
There is no such thing as a stupid question if it's sincere. Better to
ask and risk appearing stupid than to continue on your ignorant way
and make a stupid mistake