Asunción

City · 1537 CE · in South America

Capital of Paraguay (1811–today).

Buenos Aires (1536 CE): First founded by Pedro de Mendoza in 1536 and refounded in 1580 by Juan de Garay; capital of the Viceroyalty of the Río de la Plata and of Argentina.
Bogotá (1538 CE): Capital of Gran Colombia (1822–1831), Republic of New Granada (1834–1858), Granadine Confederation (1859–1862), and 2 more.
Sucre (1538 CE): Capital of Bolivia (1825–today).
La Paz (1535 CE): Capital of Republic of Baja California (1853–1854).
Lima (1535 CE): Founded by Francisco Pizarro as the "City of Kings" in 1535, Lima became the capital of the Viceroyalty of Peru and of independent Peru.
Tunja (1539 CE): Capital of Granadine Confederation (1859–1862).
Quito (1534 CE): Capital of Republic of Ecuador (1840–today).
Thanjavur (c. 1534 CE): Capital of Nayaks of Thanjavur (1534–1673), Thanjavur Maratha Kingdom (1677–1799).
Santiago (1541 CE): Capital of Republic of Chile (1812–today).
Alba Iulia (c. 1542 CE): Capital of Eastern Hungarian Kingdom (1542–1570), Principality of Transylvania (1572–1692).

Around the same time and place

Potosí (1545 CE): Mining city founded below the silver mountain Cerro Rico, whose vast output, largely extracted through forced indigenous labour, fuelled the Spanish Empire and global trade.
Amerigo Vespucci’s voyage along the coast of Brazil (1501–1502 CE): Sailing for Portugal, the Florentine Amerigo Vespucci followed the Brazilian coast far to the south. His letters arguing that these lands were a new continent led the mapmaker Waldseemüller to name it America in 1507.
La Paz (1548 CE): Capital of Bolivia (1899–today).
Rio de Janeiro (1565 CE): Founded by the Portuguese on Guanabara Bay, Rio became the capital of colonial Brazil in 1763 and of the Brazilian empire and republic until 1960.