We update the study of the polarisation of J/psi produced in proton-proton collisions at RHIC at sqrt(s)=200 GeV using the QCD-based Colour-Singlet Model (CSM), including next-to-leading order partonic matrix elements from gluon and light quark fusion and leading-order contributions from charm-quark initiated processes. To do so, we also evaluate the corresponding cross section differential in P_T which agrees qualitatively with the measurements of PHENIX in the central and forward regions at

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... w P_T -- for instance below 2 GeV --, while emphasising the need for Initial State Radiation (ISR) resummation. At mid P_T, we also compare the measurements from PHENIX and STAR with the same evaluation complemented with the dominant alphaS^5 contributions (NNLO*). We find a reasonable agreement with the data. Regarding the polarisation, as shown for previous studies at larger sqrt(s) and P_T, the polarisation pattern from gluon and light quark fusion in the helicity frame is drastically modified at NLO and is shown to be increasingly longitudinal. The yield from charm-gluon fusion is found to be slightly transversally polarised. Combining both these contributions with a data-driven range for the polarisation of J/psi from chi_c, we eventually provide an evaluation of the polarisation of the prompt J/psi yield which is in a good agreement with the experimental data from PHENIX both in the central and forward regions.