Read that temperature can cause modulation and internalization of surface antigens and this can affect fluorescent intensity in flow cytometry measurements.
I have kept cells on ice for 2 hours without any obvious affects. The cold will greatly slow any exchange of surface. I have performed actual experiments looking for internalization, but only took those out 30 minutes. I saw next to no internalization when the cells were on ice. This was for viral infectivity assays.
I believe you will not have any problem if you keep your cells on ice and use ice cold buffers. Temperatures below 4oC slows down a lot or stops protein traffic, shedding and protease activity.
If you are looking for surface markers, there could be shedding and/or internalisation of the surface molecules which would effect your results. It is best to keep them at room temperature (17-25 degrees) before staining. After staining, if you need to wait before you run the samples, use a fixation method to fixate the samples and keep them in the refrigerator (+/- 4 degrees).
EDTA will disrupt the calcium dependent internalisation as adding EDTA below the membane / phospholipid freezing point of the membrane (usually around 8-12oC) will extract intracellular calcium as well. This us why you best keep the cells somewhere around 15oC but ice water stifles the internalisation quite well. However you have to consider longer staining times when working on ice due to antibody binding kinetics and binding interactions which depend on the raft-formation (coming together of receptors in the membrane to from the receptor complex) will not work on frozen membranes
The EDTA is not always sufficient to detach cells so you should look at your harvesting efficiency anyhow to check if you need trypsin or acutase, but the enzymatic methods risk shaving bits of your cells as well.
The answer to your question is highly dependent on cell type. Many blood cells - including adherent monocytes - are perfectly happy to be in PBS/.1%BSA on ice, with or without azide for staining. Other cell types are less tolerant of extended times on ice or in the cold without clumping or undergoing apoptosis. An hour or two on ice should be fine for most cell types, but you probably should run a time course test to be sure.