May be, yes, but usually you need additional informations from other spectral tools

1D proton alone is insufficient. You need to use a technique that allows correlations between residues. HMBC or NOESY/ROESY would work.

You may be able to count number of peptide groups HN protons (8 for cyclic, 7 for acyclic octapeptide.  You may look for difference in 1H spectrum in aprotic (DMSO) vs  protic (H2O) solvent.  Terminus NH2 groups of peptides are much more exchangeable with water, and usually unobserved in a protic solvent..

It's probably worth the extra time to use 2D methods in this case, COSY, HMBC and HSQC would significantly reduce the time spent interpreting a 1D spectrum.

Use of NOE experiment using to two important peaks of ID NMR may provide some information about cyclic or acyclic nature.

You may go fro ID (1H, C13 and DEPT135) and 2D NMR (COSY, HSQC ,HMBC, ROESY AND NOESY. By this way you can interpret the structure and may know the actual result.

Not NMR based, but if you're monitoring head-to-tail cyclization a Kaiser test is a good control; depending on the sequence.  

Mass spec is the way to go. If you want to use 1D NMR, then may be compare the spectra of the linear vs cyclic peptide. You might pick up differences.

You can perfom TOCSY to see if the are some NH missed and ROESY to check some spatial connectivities..