Depends what you mean by no replicates. If you mean you have, for example, a bunch of different patients, some with disease and some without, but you only have one sample from each, then you do have replicates - each patient counts as a replicate.

If you mean you have a cell line and you have treated and untreated cells and only one sample from each, then you do not have replicates.

With no replicates, the only IDR based analysis you can do is to check for internal consistency by generating pseudo-replicates and selfing for self consistency.

Without replicates you basically can't do differential analysis of your ATAC-seq. The best you would be able to do is to get your set of peaks from your unreplicated condition and ask which of those peaks are present in neither of the peak sets from any of the replicates of the replicated set, and ask which peaks that are reproducible in the replicated condition are not present in the unreplicated condition.

For the unreplicated experiment you just won't be using IDR.

I assume by "differential expression" you mean integrating your ATAC-seq data with RNA-seq data. At the end of the day that's the same regardless of whether the ATAC-seq has replicates or not. You're generally looking for DE genes downstream of or overlapping ATAC-seq peaks. You could do differential peak calling between the conditions and look at DE genes in the resulting peaks, that'd be the simplest route.