In short, yes. All you really need to make a star map, besides lots of patience and good eyesight, is some way to measure the position of a star in the sky. So you need to be able to measure its altitude and its azimuth.

A star's altitude is its height in the sky, measured as the angle between the star and the closest point on the horizon. Thus it varies between 90 degrees (directly overhead) and 0 degrees (on the horizon).

The azimuth is the direction of that point on the horizon, usually measured as an angle from North. So 0 degrees would be North, 90 degrees would be East, 180 degrees would be South, 270 degrees would be West.

Today, we would likely use a compass to measure the azimuth. Compasses were around in the period you asked about but they probably count as "elaborate equipment". Instead, your would-be astronomer could find East by observing the sunrise on an equinox.

Alternatively, he could find North by observing the North Star.

But would someone in the 13th century know about equinoxes and the North Star? Common sense would suggest that he would: by that time, the North Star had been used for navigation for centuries, while equinoxes have been known about for millennia. However, we need not rely on common sense. In 1175, Gerard of Cremona translated Ptolemy's Almagest from Arabic into Latin, which had been translated from the original Greek into Arabic around three hundred years earlier, and which had never been forgotten in Byzantium anyway. The Almagest contains all that you could ever wish to know about equinoxes and much more besides, so we can confidently say that an astronomer in 13th century Europe or the Middle East would know all about this.

(I'm afraid I don't know about the Far East but, given how basic this astronomy is, I would be very surprised if Chinese astronomers had not written this down long ago.)

The Almagest does not contain useful information about the North Star, as the precession of the equinoxes means that there was no Pole Star in the north when Ptolemy was writing (but the Almagest does contain useful information about the precession of the equinoxes).

However, we can show that 13th century European navigators had been using the North Star for at least three hundred years thanks to a 10th century Anglo-Saxon manuscript which contains these sentences:

Twegen steorran standaþ eac stille, an on suð dæle, oðer on norð dæle, þa synd on leden axis gehatene. þone suðran steorran we he geseoð næfre; þone norðran we geseoð; þone hatað menn scip steorra.

Which Thomas Cockayne (1864, p. 271) translates as:

Two stars also stand still, one in the south part, another in the north part, which in Latin are hight axis. The southern star we never see; the northern we see; men hight it the ship star.

(hight is an old word for call or name).

Now we must measure the altitude. Mediaeval astronomers often used an astrolabe but that would definitely count as "elaborate equipment". However, a simple wooden quadrant would be easy to make. This is just a quarter circle with degrees marked around the edge. Here is a picture of Ptolemy using one.

Would a 13th century astronomer know how to make and use a quadrant? Absolutely, Ptolemy describes one in the Almagest. Furthermore, Johannes de Sacrobosco wrote the Tractatus de quadrante sometime in the mid-13th century, which describes how to construct and use a quadrant.

Again, I'm afraid I don't know about the Far East but would be surprised if they did not have something similar.

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Cockayne, T. (1864) Leechdoms, wortcunning, and starcraft of early England : Being a collection of documents, for the most part never before printed, illustrating the history of science in this country before the Norman conquest, London, Longman. Available online.

Knorr, W. (1997) 'Sacrobosco's Quadrans: Date and Sources', Journal for the History of Astronomy, vol. 28, no. 3, pp. 187-222. Available online.

Ptolemaeus (1515) Almagestum, Venice, Petrus Lichtenstein. Available online.