What are the oldest written documents in a Papuan language (ie a non-Austronesian language of the New Guinea region?) I'm not totally sure, but a strong candidate has to be the court records of Ternate. The islands of Ternate and Tidore in eastern Indonesia speak two closely related languages belonging to the non-Austronesian North Halmaheran family. They have been writing using the Jawi Arabic script since at least the 1500s; in fact, some of the earliest surviving Malay manuscripts are letters from the sultan of Ternate from about 1521.
Recently I came across an 1890 book on Ternate online: Ternate: The Residency and its Sultanate. The book includes a brief introduction to the language and a word list; it also gives reproductions of several manuscripts whose originals date back to the mid-1800s, along with translations. So if you want to try your hand at deciphering them, or just see what a Papuan language looks like in Arabic script, have a look! The page I've linked to (Arabic interpolation de-italicised) starts:
ma-dero toma hijratu-nnabiyy ṣallī `alayhi wa-sallim nyonyohi pariama calamoi si-raturomdidi si-nyagisio si-rara, tahun alif, toma-arah Sawal, i-fani futu nyagimoi si-tomodi, malam Jumaatu...
"In the year Alif of the Moslem era 1296, during the month of Sawal, on a Thursday night, the seventeenth night of the moon..."